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COMPLETE INTERSECTIONS IN SPHERICAL VARIETIES

KIUMARS KAVEH AND A. G. KHOVANSKII

Dedicated to Joseph Bernstein on the occasion of his 70th birthday

Abstract. Let G be a complex reductive algebraic group. We study complete inter-sections in a spherical homogeneous space G/H defined by a generic collection of sec-tions from G-invariant linear systems. Whenever nonempty, all such complete intersec-tions are smooth varieties. We compute their arithmetic genus as well as some of theirhp,0 numbers. The answers are given in terms of the moment polytopes and Newton-Okounkov polytopes associated to G-invariant linear systems. We also give a necessaryand sufficient condition on a collection of linear systems so that the corresponding genericcomplete intersection is nonempty. This criterion applies to arbitrary quasi-projectivevarieties (i.e. not necessarily spherical homogeneous spaces). When the spherical ho-mogeneous space under consideration is a complex torus (C∗)n, our results specialize towell-known results from the Newton polyhedra theory and toric varieties.

Contents

1. Introduction 12. Transversality and complete intersections in general varieties 53. Complete intersections in toric varieties 124. Virtual polytopes 145. Complete intersections in spherical varieties 176. Examples 30References 35

1. Introduction

The main objective of the present paper is to study complete intersections in a sphericalhomogeneous space G/H where G is a complex connected reductive algebraic group. Wecompute the arithmetic genus as well as many of the hp,0 numbers of a generic completeintersection in G/H . Our results generalize the similar results from the Newton polyhedratheory and toric varieties obtained in [Khovanskii77, Khovanskii78, Khovanskii15].

We first get a convex geometric formula for the Euler characteristic of a G-linearized linebundle over a projective spherical variety. This then allows us to represent the arithmeticgenus of a complete intersection also in terms of convex geometric data. In many cases of

Date: June 11, 2015.2010 Mathematics Subject Classification. Primary: 14M27; Secondary: 14M10.Key words and phrases. Arithmetic and geometric genus, complete intersection, spherical variety, mo-

ment polytope, Newton-Okounkov polytope, virtual polytope.The first author is partially supported by a National Science Foundation Grant (Grant ID: 1200581).The second author is partially supported by the Canadian Grant No. 156833-12.

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interest our formula is quite computable (Section 6). Our approach is based on the notionof virtual polytope as developed in [Khovanskii-Pukhlikov93]. Moreover, we use Newton-Okounkov bodies/polytopes associated to linear systems ([Okounkov97, Alexeev-Brion04,Kaveh-Khovanskii12b]) as well as string polytopes (in particular Gelfand-Zetlin polytopes)associated to irreducible representations of G ([Littelmann98, Berenstein-Zelevinsky01] and[Kaveh]).

Let G be a reductive algebraic group. A variety X with an action of G is called sphericalif a Borel subgroup of G has a dense open orbit. Spherical varieties are generalizations oftoric varieties (where G = (C∗)n is a torus) on one hand and the flag varieties G/P onthe other hand. Similar to toric varieties, geometry of spherical varieties and their orbitstructure can be read off from combinatorial and convex geometric data of fans and convexpolytopes.

We begin by recalling results about complete intersections in a torus (see also Section3). Let A1, . . . , Ak be finite subsets of Zn where k ≤ n. For each i = 1, . . . , k let fi(x) =∑

α∈Aici,αx

α be a Laurent polynomial in C[x±11 , . . . , x±1

n ] with generic coefficients ci,α. Let

Xk = {x ∈ (C∗)n | f1(x) = · · · = fk(x) = 0}

be the complete intersection in the torus (C∗)n defined by the fi. For each i = 1, . . . , k, let∆i denote the convex hull of Ai. It is an integral convex polytope in Rn.

In general a generic complete intersection Xk in (C∗)n may be empty. A beautiful resultof David Bernstein 1 gives a necessary and sufficient condition for Xk to be nonempty, interms of the dimensions of the polytopes ∆i and their Minkowski sums. It relies on atheorem of Minkowski which gives a necessary and sufficient condition for the mixed volumeof n convex polytopes to be nonzero (see [Khovanskii15]).

We recall that the arithmetic genus of a smooth complete variety Z of dimension d is

by definition χ(Z) =∑d

i=0(−1)ihi,0(Z). The geometric genus pg(Z) is the number hd,0(Z),i.e. the dimension of the space of holomorphic top forms. One knows that hp,0 numbers arebirational invariants and hence the hp,0 numbers and the arithmetic and geometric genuscan be defined for non-smooth and non-complete varieties as well.

In [Khovanskii78] the following formula for the arithmetic genus of Xk is proved. Itcomputes the arithmetic genus in terms of the number of integral points in the relativeinterior of ∆i and their Minkowski sums:

(1) χ(Xk) = 1−∑

i1

N ′(∆i1 ) +∑

i1<i2

N ′(∆i1 +∆i2)− · · ·+ (−1)kN ′(∆1 + · · ·+∆k),

where for a polytope ∆, N ′(∆) denotes the number of integral points in the interior of ∆times (−1)dim(∆). Here the interior is with respect to the topology of the affine span of∆. Moreover, if all the polytopes ∆i have full dimension n, all the hi,0(Xk) are 0 excepth0,0(Xk) = 1 and hn−k,0(Xk) which can be computed from (1). In fact, the condition thatthe polytopes have full dimension can be substitute with a weaker condition (see Corollary3.10).

As above let G be a complex connected reductive algebraic group. Let G/H be a sphericalhomogeneous space of dimension n. Let E1, . . . , Ek, k ≤ n, be G-linearized globally generatedline bundles on G/H . For each i = 1, . . . , k, let Ei ⊂ H0(G/H, Ei) be a nonzero G-invariant

1David Bernstein is the younger brother of Joseph Bernstein. He discovered the famous formula for thenumber of solutions in (C∗)n of a generic system of n polynomial equations with fixed Newton polytopes[Bernstein75]. This amazing formula inspired much activity that eventually lead to the creation of Newtonpolyhedra theory and the theory of Newton-Okounkov bodies.

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linear system. That is, Ei is a nonzero finite dimensional linear subspace of H0(G/H, Ei)and stable under the action of G. Also for i = 1, . . . , k, let fi be a generic section from thelinear system Ei. We are interested in a generic complete intersection:

Xk = {x ∈ G/H | f1(x) = · · · = fk(x) = 0}.

Let E be a G-linearized line bundle on G/H and E ⊂ H0(G/H, E) a nonzero G-invariantlinear system. Generalizing the notion of Newton polytope of a Laurent polynomial, toE one associates two polytopes: the moment polytope ∆(E) and the Newton-Okounkov

polytope ∆(E) (see Section 5.4). The moment polytope ∆(E) is defined as:

(2) ∆(E) =⋃

m>0

{λ/m | Vλ appears in Em}.

Here Em ⊂ H0(G/H, E⊗m) is the linear system spanned by all the products of m elementsfrom E and Em is the completion of Em as a linear system. The moment polytope ∆(E)contains asymptotic information about the highest weights appearing in the complete linearsystems Em for large m. The name moment polytope comes from symplectic geometrysince it coincides with the moment polytope in the sense of Hamiltonian group actions (seeRemark 5.8). In the context of reductive group actions on varieties, the notion of momentpolytope, as defined in (2), goes back to M. Brion ([Brion87]).

The Newton-Okounkov polytope ∆(E) is a polytope fibered over the moment polytope∆(E) with string polytopes as fibers (see Section 5.3). It has the property that for eachm > 0, the dimension of the complete linear system Em is equal to the number of integralpoints in the dilated polytope m∆(E). From this it follows that the number of intersections

of n generic hypersurfaces in the linear system E is equal to n!vol(∆(E)).The definition of Newton-Okounkov polytope of a spherical variety goes back to A. Ok-

ounkov for when G is a classical group (see [Okounkov97]) and V. Alexeev and M. Brion for ageneral reductive group (see [Alexeev-Brion04]). It was generalized to arbitrary G-varietiesin [Kaveh-Khovanskii12b].

The main results of the paper are as follows:Generalizing the Bernstein’s theorem, we give a necessary and sufficient condition for Xk

to be nonempty. The conditions are in terms of the dimensions of the Newton-Okounkovpolytopes of the Ei and the products of the Ei (see Theorem 5.27). The conditions can alsobe formulated only in terms of the moment polytopes. In fact, the necessary and sufficientconditions for the nonemptiness of a generic complete intersection can be formulated toapply to an arbitrary quasi-projective variety (see Theorems 2.14 and 2.19).

Moreover, whenever a generic complete intersection Xk is not empty, we give a formulafor its genus. To G/H there corresponds a sublattice in Zn and to E there corresponds

an integral point α ∈ Zn. For a polytope ∆ we count the number of points in ∆ lying inthe sublattice shifted by α. We denote by N ′(∆, α) the number of points in the sublattice

shifted by α which lie in the interior of ∆ times (−1)dim(∆) (as before the interior is with

respect to the topology of the affine span of ∆).

Theorem 1 (Theorem 5.29). The arithmetic genus χ(Xk) is given by:

(3) χ(Xk) = 1−∑

i1

N ′(∆i1(Ei1 ), αi1) +∑

i1<i2

N ′(∆(Ei1Ei2), αi1 + αi2)− · · ·

+ (−1)kN ′(∆(E1 · · ·Ek), α1 + · · ·+ αk).

Again the above can also be formulated only in terms of the moment polytopes.3

Finally we give estimates for many of the hi,0 numbers of Xk (Theorem 5.32). In partic-

ular when all the polytopes ∆(Ei) have full dimension we have the following:

Theorem 2 (Corollary 5.33). With notation as above, suppose all the polytopes ∆(Ei),i = 1, . . . , k, have full dimension equal to n = dim(G/H). Then for any 0 ≤ p < n− k:

hp,0(Xk) =

{

1, p = 0

0 p 6= 0.

Moreover, hn−k,0(Xk) can be computed from (3).

In fact, Corollary 5.33 is a slightly stronger version of the above theorem.In the last section (Section 6) we consider three important classes of spherical varieties

(aside from toric varieties): (1) horospherical varieties, (2) group embeddings, and (3) flagvarieties. In particular our formula give very computable formulae for the following concreteexamples of complete intersections in spherical homogeneous spaces:

• (A horospherical example) Let V be a finite dimensional G-module. Let v1, . . . , vsbe highest weight vectors of V with highest weights λ1, . . . , λs respectively. Putv = v1 + · · ·+ vs and let X be the closure of the G-orbit of v in V . It is an affinespherical subvariety of V . Let L be the linear subspace of C[X ] consisting of linear

functions in V ∗ restricted to X . Let ∆ = conv{λi | i = 1, . . . , s}. Also let ∆ denotethe corresponding Newton-Okounkov polytope. Let f be a generic element in Ldefining a hypersurface Hf = {x ∈ X | f(x) = 0}. Then the geometric genus of

Hf is equal to the number of integral points in the interior of the polytope ∆ (seeSection 6.1).

• (A group example) Let π : G → GL(V ) be a finite dimensional faithful represen-tation of G. Let H be a hypersurface in G defined by f = 0 where f is a genericmatrix element of π. Then the the geometric genus of H is given by the number ofpoints in the interior of the polytope ∆π (see Section 6.2).

• (A flag variety example) Let G = GL(n,C). Let Lλ be the G-line bundle on thevariety of complete flags associated to a dominant weight λ. If H is a generic divisorof Lλ then the geometric genus of H is equal to the number of integral points inRn(n−1)/2 lying in the interior of the Gelfand-Zetlin polytope ∆GZ(λ). In otherwords, the number of integral points in Rn(n−1)/2 which satisfy the inequalities in(34) where all the inequalities are strict (see Section 6.3).

The second author would like to emphasize that he enjoyed a great amount of supportfrom Joseph Bernstein during his works [Khovanskii77, Khovanskii78] and his support playedan important role in completion of these papers.

In the late 70’s the second author stated and widely advertised the problem of extendingthe results in [Khovanskii77, Khovanskii78] about geometry of complete intersections in atorus (C∗)n to other reductive groups (see Section 3). The first result in this direction wasobtained by B. Kazarnovskii ([Kazarnovskii87]) who computed the number of points in azero dimensional complete intersection in any reductive group (the answer is expressed asthe integral of a certain polynomial over an associated moment polytope). Around the sametime, M. Brion generalized Kazarnovskii’s result to a zero dimensional complete intersectionin any spherical variety ([Brion89]). Much later the first author showed ([Kaveh04]) thatthe straightforward generalization of the formula for the (topological) Euler characteristicof a complete intersection in a torus (C∗)n to a reductive group fails and such a formulashould be more complicated than in the torus case. The corresponding formula was soon

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found by V. Kiritchenko ([Kiritchenko06, Kiritchenko07]). Her unexpected and beautifulresult was at the same time slightly disappointing: the formula turns out to be unavoidablytoo complicated. This somewhat reduced the hope and suggested that perhaps extensionsof formulae for other geometric invariants from the torus case to the reductive case may betoo complicated. Nevertheless, in the present paper we give formulae for the arithmetic andgeometric genus of complete intersections in a spherical homogeneous space, exactly extend-ing the similar formulae for complete intersections in a torus (C∗)n in terms of the numberof integral points in certain associated polytopes (namely Newton-Okounkov polytopes).

The results of this paper use basic facts about the theory of virtual polytopes and convexchains as developed in [Khovanskii-Pukhlikov93]. The second author would like to point outthat A. Pukhlikov and him arrived at these ideas thinking about the Euler characteritic ofT -linearized line bundles on toric varieties.

Beside the techniques from [Khovanskii77, Khovanskii78], our results strongly rely on theresults of Michel Brion on the cohomology of G-line bundles on projective spherical varieties(Theorem 5.4), as well as the equivariant resolution of singularities of spherical varieties.

Acknowledgement: We would like to thank Michel Brion for telling us about some refer-ences regarding moment polytopes of G-varieties.

2. Transversality and complete intersections in general varieties

In this section we discuss some results on transversality and complete intersections ingeneral varieties. We will use them later in Section 5.8 to prove our main results aboutcomplete intersection in a spherical homogenous space.

2.1. Stratifications and complete intersections. Let X be a complex quasi-projectivealgebraic variety. A finite collection {Yi} of its quasi-projective subvarieties is called astratification of X , and each Yi is a stratum, if the following conditions hold: (1) The unionof all the strata is X . (2) The intersection of any two different strata is empty. (3) Eachstratum is a smooth quasi-projective variety.

Our definition of stratification, which suffices for our purposes, requires only mild as-sumptions on the strata with no condition on how a stratum approaches another stratumon its boundary (for example Whitney A and B conditions in a Whitney stratification). Onecan prove that any quasi-projective variety admits a stratification.

The following is easy to prove:

Lemma 2.1. Let X be an irreducible quasi-projective n-dimensional variety. Then anystratification {Yi} of X has exactly one n-dimensional stratum X0 and it is dense in X .

Consider a local complete intersection Z of codimension k in an n-dimensional variety Xwith a stratification {Yi}. The subvariety Z is said to be transverse to a stratum Yj of thestratification if for any point a ∈ X ∩ Yj there is a Zariski open set U ⊂ X , containing a,and a system of equations f1 = · · · = fk = 0 in U defining Z ∩U such that the differentialsof the restrictions of f1, . . . , fk to Yj are independent in the tangent space to Ta(Yj). Thesubvariety Z is transverse to the stratification {Yi} if it is transverse to all the strata of thestratification.

The following is straightforward:

Theorem 2.2. Let Z be a local complete intersection of codimension k in an n-dimensionalvariety X which is transverse to a stratification {Yi} of X. Then: (1) If Z ∩ Yj 6= ∅ the

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variety Z ∩Yj is a smooth local complete intersection of codimension k in Yj . (2) The set ofall nonempty intersections {Z ∩ Yj} form a stratification of Z. (3) If X is irreducible thenZ0 = Z ∩X0, where X0 is the stratum of dimension n, is dense in Z. (4) If X is smooththen Z is also smooth.

Let X and X be quasi-projective varieties with stratifications {Yi}, {Yj} respectively.

Definition 2.3. A morphism π : X → X respects the stratifications {Yj} and {Yi} if the

following hold: (1) π is surjective. (2) Its restriction to each stratum Yj is a surjective map

from Yj to some other stratum Yi. (3) For every x ∈ Yj the differential dπx : TxYj → Tπ(x)Yiis surjective.

The following is easy to check:

Theorem 2.4. Assume that π : X → X respects the stratifications {Yj} and {Yi}. LetZ ⊂ X be a local complete intersection of codimension k transverse to the stratification{Yi}. Then π−1(Z) = Z ⊂ X is a local complete intersection of codimension k transverse

to the stratification {Yj}.

Now let G be a complex algebraic group. In this section we consider G-varieties, i.e.varieties equipped with an algebraic action of G. Any G-variety is the union of G-orbits.We will always assume that the action of G on the variety has only finitely many orbits.It is clear that then the orbits of G give a stratification of the variety. We refer to thisstratification as a G-stratification. The following is straightforward.

Lemma 2.5. Let π : X → X be a G-equivariant surjective morphism of G-varieties X andX. Then π respects the G-stratifications of X and X.

Proof. Consider a point a ∈ X and its image b = π(a) ∈ X . Because π is G-equivariant itmaps the G-orbit of the point a onto the G-orbit of the point b. A tangent vector ξ2 at bto the orbit of b is a velocity vector of b under an action of some one-parameter subgroupG1 ⊂ G. The vector ξ2 is the image under dπ of the velocity vector ξ1 at a of the action ofG1 on X. This finishes the proof. �

For the sake of completeness we recall some basic theorems about transversality. First werecall the Bertini-Sard theorem. It is an algebraic version of the classical theorem of Sardon critical values of smooth maps on manifolds.

Theorem 2.6 (Bertini-Sard theorem). Let F : U → Ck be a morphism from a smoothalgebraic variety U to Ck and let Σ ⊂ Ck be the set of critical values of F . Then Σ is asemi-algebraic subset of Ck of codimension at least one.

We will say that some property holds for a generic point of an irreducible algebraic varietyT if there is a proper closed algebraic subset Σ ⊂ T such that the property holds for allthe points in X \ Σ (or equivalently if there is a semialgebraic set of codimension at leastone such that the property holds in its complement). In the notation of the Bertini-Sardtheorem, the set F (U) ⊂ Ck is semialgebraic. Therefore according to the theorem a genericpoint of Ck is not a critical value of F , or equivalently, a generic point of Ck is a regularvalue of F .

Lemma 2.7. Suppose T is an irreducible variety. Let Σ ⊂ T be a semi-algebraic subset.Then either x ∈ Σ holds for generic points of T , or x /∈ Σ holds for generic points of T .

Proof. If dim(Σ) = dim(T ) then dim(T \ Σ) < dim(T ). If dim(Σ) < dim(T ) then dim(T \Σ) = dim(T ). This proves the lemma. �

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Let L1, . . . ,Lk be line bundles on X , and for i = 1, . . . , k let Ek ⊂ H0(X,Li) be a finitedimensional linear subspace of sections. Let E denote the k-fold product E1×· · ·×Ek. Thefollowing is an immediate corollary of Lemma 2.7:

Lemma 2.8. Let Zg ⊂ X be the subvariety defined by g1 = · · · = gk = 0 for someg = (g1, . . . , gk) ∈ E. Then either Zg is empty for generic g ∈ E, or Zg is nonempty forgeneric g ∈ E.

With notation as in Lemma 2.8 we have the following version of the Thom transversalitytheorem. It is a corollary of the Bertini-Sard theorem. We skip the details.

Theorem 2.9 (A version of Thom’s transversality theorem). Let X be a quasi-projectivevariety equipped with a stratification {Yi}. Assume that the following hold: (1) The linearsystems E1, . . . , Ek are base point free, and (2) Zg 6= ∅ for generic g ∈ E = E1 × · · · × Ek.Then, for generic g ∈ E, the subvariety Zg is a local complete intersection of codimensionk which is transverse to {Yi}.

2.2. When is a generic complete intersection nonempty? Suppose we are given klinear systems on a variety. In this section we give a necessary and sufficient condition fora generic complete intersections from these linear systems to be nonempty. We would needthe notion of the Kodaira map of a linear system which we briefly explain below:

As above let X be an n-dimensional quasi-projective variety. Let E be a linear system onX , that is, a finite dimensional linear subspace of global sections of a line bundle L on X .Assume that E is base point free. One can then define a morphism ΦE : X → P(E∗) calledthe Kodaira map of E. It is defined as follows: ΦE(x) is the point in the projective spaceP(E∗) represented by the hyperplane Hx in E consisting of all the sections which vanish atx. We denote the closure of the image of ΦE by YE . It is a projective subvariety of P(E∗).The following is easy to prove from the definition of the Kodaira map:

Lemma 2.10. For a, b ∈ X we have ΦE(a) = ΦE(b), if and only if the sets {g ∈ E | g(a) =0} and {g ∈ E | g(b) = 0} coincide.

The notion of a Kodaira map is very useful in the theory of Newton-Okounkov bodies.One usually assumes that the linear system under consideration is large enough so that theKodaira map ΦE is an isomorphism (or at least a birational isomorphism) between X andYE . In this section we relax this and would work with general base point free linear systemsE such that YE can have smaller dimension than that of X .

Let L1, . . . ,Lk be a collection of globally generated line bundles on X . For i = 1, . . . , klet Ei ⊂ H0(X,Li) be a finite dimensional subspaces of global sections of Li without basepoint. We will use the following notation: I denotes the set of indices {1, . . . , k} andJ = {i1, . . . , ij} is a nonempty subset of I. We write LJ for the line bundle Li1 ⊗ · · · ⊗ Lij

and EJ is the subspace of H0(X,LJ) spanned by all the tensor products gi1 ⊗ · · · ⊗ gij ,where giℓ is a section of Eiℓ for ℓ = 1, . . . , j. We will denote the Kodaira map of the linearsystem EJ simply by ΦJ : X → P(E∗

J). We have the following extension of Lemma 2.10:

Lemma 2.11. For a, b ∈ X we have ΦI(a) = ΦI(b) if and only if for every i ∈ I the sets{gi ∈ Ei | gi(a) = 0} and {gi ∈ Ei | gi(b) = 0} coincide.

Proof. Let us prove that if ΦI(a) = ΦI(b) and gi(a) = 0 for some gi ∈ Ei then gi(b) = 0. Forevery j 6= i fix a section fj ∈ Ej such that fj(a) 6= 0 and fj(b) 6= 0. For any gi ∈ Ei considerthe section φ = f1 ⊗ · · · ⊗ fi−1 ⊗ gi ⊗ fi+1 ⊗ · · · ⊗ fk ∈ EI . By Lemma 2.10 the conditionsφ(a) = 0 and φ(b) = 0 are equivalent. So the equations gi(a) = 0 and gi(b) = 0 on a section

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gi ∈ Ei are equivalent. Conversely, assume that for every i the equations gi(a) = 0 andgi(b) = 0 for gi ∈ Ei are equivalent. Represent each linear space Ei in the form E0

i ⊕ E1i

where every section from E0i vanishes at the points a and b and the one-dimensional subspace

E1i is spanned by a section si not vanishing at a and b. The linear space EI is a sum of

2k subspaces En1,...,nk

I where the sum is taken over all 2k k-tuples (n1, . . . , nk) of indexesni = 0, 1, and each linear space En1,...,nk

I is spanned by the tensor products f1 ⊗ · · · ⊗ fkwhere fi ∈ E0

i if ni = 0 and fi = si if ni = 1. The sections belonging to each summand but

not to the one-dimensional subspace E1,...,1I spanned by the section s1⊗ · · ·⊗ sk vanish at a

and b. So the conditions φ(a) = 0 and φ(b) = 0 for φ ∈ EI are equivalent. By Lemma 2.10we have ΦI(a) = ΦI(b) which finishes the proof. �

Definition 2.12. With notation as above, we define the defect d(J) of a subset J ⊂ I ={1, . . . , k} to be the number:

d(J) = τJ − |J |,

where τJ is the dimension of YJ , the closure of the image of X under the Kodaira map ΦJ

and |J | is the number of elements in J .

Definition 2.13. With notation as above, we say that the linear systems E1, . . . , Ek areindependent if any subset J ⊂ I = {1, . . . , k} has nonnegative defect.

Theorem 2.14 (Necessary condition for a generic complete intersection to be nonempty).Let E1, . . . , Ek be base point free linear systems. Suppose Zg ⊂ X is nonempty for a genericchoice of g ∈ E = E1×· · ·×Ek. Then E1, . . . , Ek are independent (in the sense of Definition2.13).

Proof. Assume that the set J = {i1, . . . , ij} has a negative defect d(J) with respect tothe collection of linear systems E1, . . . , Ek. Let us show that for a generic choice of gJ =(gi1 , . . . , gik) ∈ Ei1 × · · · × Eij , the subvariety ZgJ

⊂ X defined by the equations gi1 =· · · = gik = 0 is empty. Suppose this is not the case. Fix a stratification of X with thelargest stratum X0. Then by Theorem 2.9 the variety ZgJ

is a local complete intersectionof dimension n − |J | and the intersection Z0

gJ= ZgJ

∩ X0 is nonempty and smooth . Let

a be a point in Z0gJ. By Lemma 2.11 the smooth variety Z0

gJshould contain the set of all

points x ∈ X0 such that ΦJ(a) = ΦJ (x). The dimension of the set Ka = Φ−1J (ΦJ (a)) ∩X0

is greater than or equal to n − τJ . Indeed, ΦJ restricted to X0 is a surjective morphismfrom the smooth variety X0 of dimension n to the variety ΦJ (X0) of dimension τJ . Butd(J) = τJ − |J | < 0 so n − τJ > n − |J | which is impossible because Ka ⊂ Z0

gJ. The

contradiction proves that ZgJis empty. This shows that Zg is empty as well. �

The rest of this section is devoted to proving the converse of Theorem 2.14 (Theorem2.19).

First we introduce a foliation on X using the Kodaira map of a linear system. Let E bea base point free finite dimensional linear subspace of global sections of a line bundle L onX . Let ΦE : X → P(E∗) be the corresponding Kodaira map. Below we use the followingnotation: S is the singular locus of the variety X . The number τE is the dimension of YE ,the closure of the image of the Kodaira map ΦE . The set SE is the singular locus of YEand U is the Zariski open set X \ (S ∪Φ−1

E (SE)) in X . Finally, Σc ⊂ U is the set of criticalpoints of ΦE restricted to U ⊂ X .

Definition 2.15. Let a ∈ U \ Σc. Let F (a) be the subspace of the tangent space TaU

defined by the linear equations dga = 0 for all g ∈ E. The collection of subspaces F (a)

defines an (n− τE)-dimensional distribution F on the Zariski open set U \ Σc in X .8

The next lemma is a corollary of the Implicit Function Theorem.

Lemma 2.16. The foliation F in U \ Σc is completely integrable. Its leaves are connectedcomponents of the preimages under the Kodaira map ΦE : U \ Σc → YE of the points inΦE(U \Σc).

Take a point a ∈ U \Σc and a section g ∈ E such that g(a) = 0 and dg(a) 6= 0. Let H bethe hyperplane in the tangent space TaU defined by dg = 0. The following is straightforward:

Lemma 2.17. 1) The hyperplane H contains the (n−τE)-plane F (a). 2) For any hyperplaneH ⊂ TaU which contains the plane F (a) there is a section g ∈ E such that g(a) = 0 and His defined by the equation dga = 0.

We will need a linear algebra statement about transversality of a collection of hyper-planes. Let F1, . . . , Fk ⊂ T be subspaces in a vector space T . For any nonempty setJ = {i1, . . . , ij} ⊂ {1, . . . , k} let FJ denote the the subspace Fi1 ∩ · · · ∩ Fij . The fol-lowing theorem gives a necessary and sufficient condition for the existence of hyperplanesH1, . . . , Hk ⊂ T such that: (1) Fi ⊂ Hi for 1 ≤ i ≤ k, and (2) the hyperplanes H1, . . . , Hk

are mutually transverse.

Theorem 2.18. The hyperplanes H1, . . . , Hk satisfying the above conditions exist if andonly if for any subset J the codimension of FJ in T is greater than or equal to |J |.

Proof. First suppose the subspaces H1, . . . , Hk satisfying the above conditions exist. Forany nonempty subset J = {i1, . . . , ij} let HJ denote the subspace Hi1 ∩ · · · ∩ Hij . ThenFJ ⊂ HJ and the codimension of HJ in T is |J |. So the codimension of FJ is greater thanor equal to |J |. We prove the converse statement by induction on k. Suppose that thestatement holds for any collection of (k− 1) subspaces. Choose Hk to be a hyperplane suchthat Hk contains Fk but it does not contain any subspace FJ not inside Fk (so FJ ∩ Hk

has codimension 1 in FJ). Now we apply the induction hypothesis to the k− 1 hyperplanesF ′i = Fi ∩Hk, i = 1, . . . , k − 1, in the vector space Hk. Let us verify that these satisfy the

conditions in the theorem. For any subset J ⊂ {1, . . . , k} let J∗ = J \{k}. If FJ∗ ⊂ Fk thenFJ∗ = FJ∗ ∩ Fk = FJ where J = J∗ ∩ {k}. The codimension of FJ∗ ∩ Hk in Hk is equalto the codimension of FJ in T minus 1. By the assumption, codim(FJ )− 1 is greater thanor equal to (|J∗|+ 1)− 1 = |J∗|. On the other hand, if FJ∗ is not contained in Fk then itscodimension in T is equal to the codimension of FJ∗ ∩Hk in Hk. Again by the assumptionin the theorem this is greater than or equal to |J∗|. Finally by the induction hypothesisthere are mutually transverse hyperplanes H ′

1, . . . , H′k−1 in Hk, such that Fi ∩Hk ⊂ H ′

i fori < k. Enlarge each H ′

i to a hyperplane Hi in T such that Hi contains Fi. The collection ofhyperplanes H1, . . . , Hk has the required properties. �

We can now prove the converse of Theorem 2.14.

Theorem 2.19 (Sufficient condition for a generic complete intersection to be nonempty).Let E1, . . . , Ek be a collection of base point free linear systems. Suppose that E1, . . . , Ek areindependent in the sense of Definition 2.13. Then a generic complete intersection Zg ⊂ X,where g ∈ E = E1 × · · ·Ek, is nonempty.

Proof. Let J = {i1, . . . , ij} be a nonempty subset of I = {1, . . . , k}. Consider the Kodairamap ΦJ associated with the space EJ . Using Lemma 2.16 one can find a smooth Zariskiopen subst U ⊂ X and integrable foliations FJ in U of codimensions τJ whose leaves areconnected components of the manifolds Φ−1

J (b) ∩ U where b ∈ ΦJ(U). Take a point a ∈ Uand let T = TaU be the tangent space to U at a. Let F1, . . . , Fk be the subspaces in T

9

tangent to the leaves of these foliations. For any nonempty J = {i1, . . . , ij} the intersection

FJ = Fi1 ∩ · · · ∩ Fij ⊂ T coincides with the subspace tangent to the leaf of the foliation FJ

passing through a (Lemma 2.11). The codimension τJ of FJ in T is d(J) + |J | ≥ |J |. Thusapplying Theorem 2.18 we can find mutually transverse hyperplanes H1 . . . , Hk ⊂ T suchthat Fi ⊂ Hi for i = 1, . . . , k. By Lemma 2.17 there are sections gi ∈ Ei, i = 1, . . . , k suchthat gi(a) = 0 and the tangent hyperplane to {gi = 0} at a is Fi. Hence the hypersurfacesgi = 0 at a neighborhood of the point a are smooth and mutually transverse. By the ImplicitFunction Theorem we then know that for a k-tuple of sections g = (g1, . . . , gk) close enoughto g the variety Zg is nonempty. Thus it is not true that for a generic g the variety Zg isempty. This finishes the proof of the theorem. �

2.3. The hp,0 numbers of a complete intersection. The material in this section aretaken from [Khovanskii15]. Let L be a line bundle on a smooth projective variety X andlet D be a divisor of L. We denote by O(X,L) or O(X,D) the germ of regular sections ofL. We denote the i-th sheaf cohomology group of X with coefficients in the sheaf O(X,L)by Hi(X,L) or Hi(X,D). We will write Hi(X) for the i-th cohomology group of the zerodivisor, that is, Hi(X) is the i-th cohomology of X with coefficients in OX , the sheaf ofgerms of regular functions. For an integer p ≥ 0, we have dim(Hp(X)) = hp,0(X), thedimension of the space of holomorphic p-forms on X . In particular, h0,0(X) is the numberof irreducible components of X and hp,0(X) = 0 for p > dim(X).

Fix a collection L1, . . . ,Lk of globally generated line bundles on X . Recall that a linebundle is globally generated if for any x ∈ X there is a global section that does not vanishat x. For each i = 1, . . . , k let fi ∈ H0(X,Li) be such that the divisor Di defined byfi = 0 is a smooth hypersurface. Moreover, assume that the divisors D1, . . . , Dk intersecttransversely. We will be interested in the local complete intersection Xk = D1 ∩ · · · ∩Dk.From transversality it follows that this intersection is a smooth subvaritey of X .

Given the dimensions of the cohomology groups:

Hi(X,L⊗m1

1 ⊗ · · · ⊗ L⊗mk

k ), m1, . . . ,mk ∈ {0,−1},

of the ambient variety X , one can obtain much information about the hp,0 numbers of thecomplete intersection Xk. In particular one can compute the arithmetic genus of Xk. Belowwe recall how this can be done ([Khovanskii78]).

Form = 1, . . . , k let Xm = D1∩· · ·∩Dm. We then have a sequence of smooth subvarietiesXk ⊂ · · · ⊂ X0 = X where each variety is a hypersurface in the next one. For a line bundleL consider the exact sequence of sheaves:

0 → O(Xm−1,L ⊗ L−1m )

i→ O(Xm−1,L)

j→ O(Xm−1,L) → 0.

Here O(Xm−1,L ⊗ L−1m ) is the sheaf on Xm−1 of regular sections of the line bundle

L⊗L−1m on Xm−1. The sheafs O(Xm−1,L) and O(Xm,L) have analogues definitions. The

sheaf O(Xm−1,L) is the trivial extension of the sheaf O(Xm,L) on Xm to a sheaf on Xm−1.The homomorphism i maps a section g of L⊗L−1

m to a section g⊗ fm where fm is a globalsection of Lm defining the divisor Dm, and the homomorphism j at a ∈ Xm maps a sectionof L on Xm−1 to its restriction to Xm, and at a point a ∈ Xm−1 \Xm the homomorphismj is trivial. The long exact sequence of the cohomology groups corresponding to the aboveexact sequence of sheaves is as follows:

(4) 0 → H0(Xm−1,L⊗ L−1m ) → H0(Xm−1,L) → H0(Xm,L) → · · · ,

(the cohomology of the sheaves O(Xm,L) and O(Xm−1,L) are canonically isomorphic).10

For a complete variety X with a sheaf F we denote the Euler characteristic of X withcoefficients in F by χ(X,F):

χ(X,F) =n∑

i=0

(−1)idim(Hi(X,F)).

In particular, we write χ(X,L) for the Euler characterisitic of X with coefficients in thesheaf O(X,L) of sections of a line bundle L.

The Euler characteristic is additive, i.e. if G is a sub-sheaf of sheaf F on X then:

χ(X,F) = χ(X,G) + χ(X,F/G),

where F/G is the quotient sheaf. The exact sequence (4) then allows us to find the Eulercharacteristic χ(Xk,L). We give the answer for the trivial bundle, that is, χ(Xk):

Theorem 2.20. The arithmetic genus χ(Xk) of the smooth variety Xk is equal to:

χ(X)−∑

i1

χ(X,L−1i ) +

∑

i1<i2

χ(X,L−1i1

⊗ L−1i2

)− · · ·+ (−1)kχ(X,⊗

1≤i≤k

L−1i ).

For a nonempty set J ⊂ {1, . . . , k} let L−1J =

⊗

i∈J L−1i .

Theorem 2.21. We have the following upper bound for the hi,0 numbers of the completeintersection Xk:

(5) hi,0(Xk) ≤ hi,0(X) +∑

J 6=∅

dim(Hi+|J|(X,L−1J )).

Proof. We can rewrite (5) as hi(Xk) ≤∑

J dim(Hi+|J|(X,L−1J )). Let L be any line bundle

on X . We will prove the following more general inequality which coincides with (5) when Lis the trivial line bundle:

(6) dim(Hi(Xk,L)) ≤∑

J

dim(Hi+|J|(X,L⊗ L−1J )).

We prove (6) by induction on k. Let j ≥ 0 and 1 ≤ m ≤ k. From the piece:

→ Hj(Xm−1,L) → Hj(Xm,L) → Hj+1(Xm−1,L⊗ L−1m ) → . . .

of the exact sequence (4) we obtain that:

(7) dim(Hj(Xm,L)) ≤ dim(Hj(Xm−1,L)) + dim(Hj+1(Xm−1,L ⊗ L−1m )).

For k = 1, the inequality (6) coincides with (7) for j = i and m = 1. Assume that (6) isproved for k − 1. For J ⊂ {1, . . . , k} let J∗ = J ∩ {1, . . . , k − 1}. Then either J = J∗ orJ = J∗ ∪ {k}. In the first case we have:

(8) |J | = |J∗| and L ⊗ L−1J = L ⊗ L−1

J∗ .

In the second case we have:

(9) |J | = |J∗|+ 1 and L⊗ L−1J = L⊗ L−1

J∗ ⊗ L−1k .

By induction hypothesis we can assume that for any line bundle L and i ≥ 0 the followinginequality holds:

(10) dim(Hi(Xk−1,L)) ≤∑

J∗

dim(Hi+|J∗|(X,L ⊗ L−1J∗ )),

11

where the summation is over all the subsets J∗ ⊂ {1, . . . , k−1}. Also by induction hypothesiswe know that for the line bundle L ⊗ L−1

k and any i+ 1 the inequality:

(11) dim(Hi+1(Xk−1,L ⊗ L−1k )) ≤

∑

J∗

dim(Hi+1+|J∗|(X,L⊗ L−1k ⊗ L−1

J∗ )),

holds. Here also the summation is taken over all subsets J∗ ⊂ {1, . . . , k − 1}. Now insteadof the numbers dim(Hi(Xk−1,L)) and dim(Hi+1(Xk−1,L⊗L−1

k )) plug the righthand sidesof (10) and (11) into (7). Using (8) and (9) we obtain the required inequality (6). Thetheorem is proved. �

We have the following direct corollary of Theorem 2.21:

Corollary 2.22. Assume that for some integer i the cohomology groups Hi+|J|(X,L−1J )

vanish for any nonempty J ⊂ {1, . . . , k}. Then hi,0(Xk) ≤ hi,0(X).

3. Complete intersections in toric varieties

The material in this section are taken from [Khovanskii78] and [Khovanskii15]. We ex-plain the results on genus of complete intersections in torus (C∗)n from [Khovanskii78]concerning the case when the Newton polytopes involved are full dimensional, as well astheir extensions in [Khovanskii15] to the case when the polytopes are not necessarily fulldimensional.

Let ∆1, . . . ,∆k be integral polytopes in Rn (i.e. with vertices in Zn). Let f1, . . . , fk bea generic k-tuple of Laurent polynomials with Newton polytopes ∆1, . . . ,∆k respectively.Let Z be the subvariety of (C∗)n defined by f1 = · · · = fk = 0.

Let X be a smooth projective toric variety which is sufficiently complete with respectto the polytopes ∆1, . . . ,∆k in the sense of [Khovanskii77]. We let D∞ denote the divisorat infinity of a generic Laurent polynomial with Newton polytope ∆ = ∆1 + · · · + ∆k onX . It is a divisor supported on the complement of the open orbit (C∗)n and hence is torusinvariant. Let L∞ be the line bundle corresponding to D∞.

Since the divisor D∞ is torus invariant one can use theory of toric varieties to computethe sheaf cohomology groups of D∞ in terms of the polytope ∆ ([Khovanskii77, Section 4]).We recall the answer below. We need a bit of notation: for a polytope ∆ we denote thenumber of integral points in ∆ by N(∆). Also N◦(∆) denotes the number of integral pointsin the interior of the polytope ∆. Here the interior is with respect to the topology of theaffine span of ∆. We also write N ′(∆) for (−1)dim(∆)N◦(∆). 2

Theorem 3.1. We have:

dim(Hi(X,L−1∞ )) =

{

0, i 6= dim(∆)

N◦(∆), i = dim(∆)

We note that if X is sufficiently complete for the polytope ∆ = ∆1 + · · · + ∆k then itis also sufficiently complete for m1∆1 + · · ·+mk∆k for any integers mi ≥ 0. In particular,the cohomology groups of the line bundles associated to m1∆1 + · · ·+mk∆k are also givenby Theorem 3.1.

Corollary 3.2. Let X be a sufficiently complete for ∆. Then the Euler characteristicχ(X,L−1

∞ ) is equal to N ′(∆).

2In [Khovanskii77, Khovanskii78] instead of our notation N(∆), N◦(∆) and N ′(∆) respectively thenotation T (∆), B+(∆) and B(∆) is used.

12

Corollary 3.3. For any smooth projective toric variety X we have h0,0(X) = 1 andhi,0(X) = 0 for i > 0.

Proof. The numbers hi,0 are birational invariants and any n-dimensional toric variety isbirationaly equivalent to CPn. For CPn the corollary is obvious. Nevertheless let us deducethe corollary from Theorem 3.1. Let ∆ = {0} be the polytope consisting of the singlepoint 0. Then dim(∆) = 0 and N◦(∆) = 1. Any smooth projective toric variety X issufficiently complete for ∆ = {0} and the divisor D∞ on X is {0}. Now apply Theorem 3.1for ∆ = {0}. �

We now use the results in Section 2.3 to give a condition for when a generic completeintersection Z is irreducible. We also prove a result about the hp,0 numbers of Z.

For an integral polytope ∆ ⊂ Rn let L∆ denote the linear subspace of Laurent polynomialsin (C∗)n spanned by all the monomials xα where α ∈ ∆ ∩ Zn. It is easy to see that L∆ hasno base point on (C∗)n. Let Φ∆ : (C∗)n → P(L∗

∆) denote its Kodaira map. One observesthat the dimension of the image of Φ∆ is equal to dim(∆).

As above, let ∆1, . . . ,∆k be integral polytopes in Rn. For each i = 1, . . . , k let Li = L∆i

be the corresponding subspace of Laurent polynomials. From the above it follows that thedefect of a subset J ⊂ {1, . . . , k} is equal to:

d(J) = dim(∆J )− |J |,

where ∆J =∑

i∈J ∆i. We call ∆1, . . . ,∆k independent if the corresponding subspacesL1, . . . , Lk are independent (see Definition 2.13). In other words, ∆1, . . . ,∆k are indepen-dent if dim(

∑

i∈J ∆i) ≥ |J | for any subset J ⊂ {1, . . . , k}.Now let

(12) f1(x) = · · · = fk(x) = 0

be a generic system of Laurent polynomials with Newton polytopes ∆1, . . . ,∆k respectivelydefining a complete intersection Z in (C∗)n. We will assume that the polytopes ∆1, . . . ,∆k

are independent. This guarantees that Z is nonempty (Theorem 2.14). As before we letX be a fixed smooth projective toric variety with (C∗)n as the open orbit whose fan is asubdivision of the normal fan of the polytope ∆ = ∆1 + · · · + ∆k (then X is sufficientlycomplete with respect to the polytopes ∆1, . . . ,∆k). Also let X∞ denote the sum of primedivisors in X which lie in the complement of the open orbit (C∗)n. It is well-known thatsince X is smooth, the divisor X∞ has normal crossings.

To apply the result in Sections 2.3 to the variety Z we need the following lemma. For aproof see [Khovanskii77].

Lemma 3.4. With notation as above, let Di be the closure of the hypersurface defined byfi = 0 in X, for i = 1, . . . , k. Then Di is a smooth hypersurface and moreover all thedivisors Di and the closures of all the (n− 1)-dimensional orbits are mutually transverse inX.

Assume that the variety Z is nonempty (i.e. the generic system (12) has solutions) andthe conditions of Lemma 3.4 hold. Then the closure of Z in X is the intersection of thesmooth divisors D1, . . . , Dk. As before, for each i let Di,∞ denote the divisor at infinityon the toric variety X associated to the polytope ∆i and let Li,∞ be its corresponding linebundle. Theorem 3.1 and the remark after it, give us the needed information about thedimensions of the cohomology groups Hi(X,L⊗m1

1,∞ ⊗ · · · ⊗ L⊗mk

k,∞ ), m1, . . . ,mk ∈ {0,−1}.13

Theorem 3.5. The arithmetic genus χ(Z) of Z, defined by a genreic system (12), is givenby:

(13) χ(X) = 1−∑

i1

N ′(∆i1) +∑

i1<i2

N ′(∆i1 +∆i2 )− · · ·+ (−1)kN ′(∆1 + · · ·+∆k).

Recall that N ′(∆) denotes (−1)dim(∆)N◦(∆) and N◦(∆) is the number of integral points inthe interior of ∆.

Proof. Theorem follows immediately from Theorem 2.20 and Corollary 3.2. �

Remark 3.6. For k = n the righthand side of the (13) is equal to n!V (∆1, . . . ,∆n) where Vdenotes the mixed volume of convex bodies ([Bernstein75]). So the Bernstein-Kushnirenkotheorem follows from Theorem 3.5 (note that if dim(Z) = 0 then χ(Z) is equal to the numberof points in Z). The formula for χ(Z) above and deduction of the Bernstein-Kushnerenkotheorem from this formula are from [Khovanskii78].

Finally we have the following theorems about the hp,0 numbers of the complete intersec-tion Z:

Theorem 3.7. For any integer i ≥ 0 the following holds:

hi,0(Z) ≤∑

{J|dim(∆J )−|J|=i and J 6=∅}

N◦(∆J ) + δi0,

where δi0 = 0 for i 6= 0 and δ00 = 1.

Proof. The theorem follows from Theorem 2.21, Theorem 3.1 and the identity hi,0(X) = δi0(see Corollary 3.3). �

Definition 3.8. Let ∆1, . . . ,∆k be a k-tuple of independent integral polytopes. We saythat a number i ≥ 0 is critical for ∆1, . . . ,∆k if there is a nonempty set J ⊂ {1, . . . , k} suchthat N◦(∆J ) > 0 and dim(∆J)− |J | = i.

Theorem 3.9. Let ∆1, . . . ,∆k be a k-tuple of independent integral polytopes.

(a) If 0 is a non-critical number for the collection of the ∆i then the variety Z definedby a generic system (12) is irreducible.

(b) If i > 0 is a non-critical number for ∆1, . . . ,∆k then hi,0(Z) = 0.

Proof. For a non-critical i the inequality in Theorem 3.7 becomes hi,0(Z) ≤ δi0. But thenumber h0,0(Z) is equal to the number of irreducible components of Z and therefore it isstrictly positive. The numbers hi,0(Z) are nonnegative. �

Theorem 3.7 implies the following improvement of a result in [Khovanskii78]:

Corollary 3.10. If all the numbers 0 ≤ i < n − k are not critical for ∆1, . . . ,∆k thenh0,0(Z) = 1, hn−k,0(Z) = (−1)(n−k)(χ(Z)− 1) and hp,0(Z) = 0 for p 6= 0 and p 6= n− k.

Proof. From Theorem 3.7 we have h0,0(Z) = 1 and hp,0(Z) = 0 for 0 < p < n − k. Thedimension of the smooth variety Z is n− k and therefore hp,0(Z) = 0 for n− k < p. �

4. Virtual polytopes

In this section we recall some basic facts from the theory of finitely additive measureson virtual polytopes developed in [Khovanskii-Pukhlikov93]. We will need them later inSection 5. This theory extends the theory of valuations on convex polyhtopes due to PeterMcMullen ([McMullen77]) and uses the integration with respect to the Euler characteristicdeveloped by Oleg Viro ([Viro88]).

14

4.1. Ring Z(Λ) of convex Λ-chains. For a fixed natural number N let 1NZn be the lattice:

1

NZn = {a/N | a ∈ Zn} ⊂ Rn.

In the rest of this section Λ ⊂ Rn denotes an additive subgroup of Rn which is eitherequal to Rn itself or is equal to 1

NZn for some natural number N . We denote by P(Λ) thecollection of all convex polytopes in Rn with vertices in Λ. We call the elements of P(Λ)the Λ-polytopes. When Λ = Rn (respectively Λ = Zn) we refer to the Λ-polytopes simply aspolytopes (respectively integral polytopes).

Definition 4.1. A convex Λ-chain (or a Λ-chain for short) is a function α : Rn → Z whichcan be represented as a finite sum α =

∑

i niχ∆i, where χY denotes the characteristic

function of a set Y , the ∆i are convex polytopes in P(Λ) and ni ∈ Z. We denote theadditive group of convex Λ-chains by Z(Λ).

Let ∆◦ be the set of interior points of a polyhedron ∆, in the topology of the affine spacespanned by ∆. It is easy to see that

χ∆◦ =∑

∆i∈Γ(∆)

(−1)dim∆iχ∆i,

where Γ(∆) is the set of all faces of ∆, including ∆ itself. Therefore if ∆ is in P(Λ) thenχ∆◦ is also in Z(Λ).

It is shown in [Khovanskii-Pukhlikov93] that the Minkowski sum of convex polytopesextends in a unique way to an opration ∗:

∗ : Z(Λ)× Z(Λ) → Z(Λ),

on the additive group of Λ-convex chains. That is, for two convex polytopes ∆1,∆2 ∈ P(Λ)we have:

χ∆1∗ χ∆2

= χ(∆1+∆2).

We call the operation ∗ the (Minkowski) multiplication of chains. The group Z(Λ) togetherwith this multiplication is a commutative ring called the ring of Λ-convex chains.

The characteristic function of the origin χ{0} is the unit in the ring Z(Λ). We call anyinvertible element of the ring of Λ-chains a Λ-virtual polytope. When Λ = Rn (respectivelyΛ = Zn) we simply call an invertible element a virtual polytope (respectively an integralvirtual polytope). It turns out that the characteristic function χ∆ of a polytope ∆ ∈ P(Λ)is invertible in the ring Z(Λ):

Theorem 4.2 (Λ-virtual polytopes). The following statements hold:

(1) For ∆ ∈ P(Λ) the multiplicative inverse χ−1∆ of χ∆ ∈ Z(Λ) is

(−1)dim∆χ(−∆◦).

Here −∆◦ is the set of interior points of −∆.(2) Each Λ-virtual polytope can be written in the form χ∆1

∗χ−1∆2

, where ∆1,∆2 ∈ P(Λ).

4.2. Integration over the Euler characteristic and multiplication in the ring Z(Λ).There is a way to define the multiplication ∗ in the ring of convex chains Z(Λ) which doesnot require representation of convex chains as linear combinations of characteristic functionsof polytopes. This definition instead uses integration with respect to the Euler characteristic(see [Viro88]). Below we explain this notion in more detail.

15

Let us say that a subset Y in a finite dimensional real vector space V is semi-convex ifY can be represented as a disjoint union

(14) Y =⋃

i

∆◦i ,

of relative interiors of finitely many convex polytopes ∆i ∈ P(Λ). By definition the Eulercharacteristic µ(Y ) of a semi-convex set Y is

∑

i(−1)dim∆i . It is known (see [Viro88]) thatµ(Y ) is well-defined, i.e. is independent of a choice of the representation (14). The Eulercharacteristic is a finitely additive measure on the collection of semi-convex sets. For closedsemi-convex sets this measure coincides with the usual Euler characteristic in topology, butfor a general semi-convex set it is not the topological Euler characteristic. One defines theintegral

∫

fdµ of a convex chain f ∈ Z(Rn) with respect to the Euler characteristic by:∫

fdµ =∑

a∈Z

aµ(f−1(a)).

(Note that f takes only finitely many values and that each level set f−1(a) is a semi-convexset.)

Theorem 4.3 (Multiplication in Z(Λ) using the Euler characteristic). The multiplicationα ∗ β of α, β ∈ Z(Λ) is equal to the convolution with respect to the integration over Eulercharacteristic of the functions α, β:

(α ∗ β)(x) =

∫

α(z)β(x − z)dµ(z).

4.3. Finitely additive measures on convex Λ-chains. A (real valued finitely additive)measure on Z(Λ) is by definition an additive homomorphism φ : Z(Λ) → R. A measure iscalled an invariant measure (respectively a polynomial measure of degree ≤ d) if for eachα ∈ Z(Λ) the function h 7→ φ(αh) on h ∈ Λ is constant (respectively is the restriction ofa polynomial of degree ≤ d to Λ ⊂ Rn). Here αh ∈ Z(Λ) is the convex chain defined byαh(x) = α(x− h) (i.e. αh is the shift of α by h).

One can check the following easy facts:

Lemma 4.4. Let φ : Z(Rn) → R be a measure. Then:

(1) For β ∈ Z(Rn) the function φβ : Z(Rn) → R defined by φβ(α) = φ(α ∗ β) is also ameasure on Z(Rn).

(2) The restriction of φ to Z(Λ) ⊂ Z(Rn) is a measure on Z(Λ).

Lemma 4.5. Let f : Rn → R be a polynomial of degree ≤ d. Then the function φ =∫

f onZ(Rn) defined by φ(α) =

∫

f(x)α(x)dx, where dx is the standard Lebesgue measure on Rn,is a polynomial measure of degree ≤ d on Z(Rn).

In particular, Lemma 4.5 states that the map ∆ 7→∫

∆ f(x)dx on the space of polytopesP(Rn) uniquely extends to a measure on virtual polytopes. This measure is the usual volumeof a polytope if f is the constant polynomial 1.

Lemma 4.6. Let f : Rn → R be a polynomial of degree ≤ d and let β ∈ Z(Rn) be a convexchain. Then the function φ on integral convex chains Z(Zn) defined by:

φ(α) =∑

x∈Zn

f(x)(β ∗ α)(x),

is a polynomial measure of degree ≤ d on integral convex chains.16

In particular Lemma 4.6 implies the following: let ∆β be a fixed (not necessarily integral)convex polytope. Consider the map φ : ∆ 7→

∑

x∈(∆+∆β)∩Zn f(x) on convex polytopes.

Then this map extends uniquely to a measure on virtual convex polytopes. If f is theconstant polynomial 1 then φ(∆) is the number of integral points in ∆ +∆β .

We now state the main statement about polynomial measures on virtual polytopes. Fixa k-tuple of Λ-virtual polytopes α1, . . . , αk ∈ Z(Λ) as well as a polynomial measure φ ofdegree ≤ d on Z(Λ). Let Φ be the function on Zk defined by:

Φ(m1, . . . ,mk) = φ(αm1

1 ∗ · · · ∗ αmk),

where (m1, . . . ,mk) ∈ Zk and αmi

i denotes the mi-th power of αi in the ring Z(Λ).

Theorem 4.7. With the above notation, the function Φ is the restriction of a polynomialof degree ≤ n+ d on Rk to Zk.

The next corollary follows from Theorem 4.7 applied to the measure in Lemma 4.6.

Corollary 4.8. Let f : Rn → R be a degree d polynomial, β a convex chain and α1, . . . , αk

a k-tuple of integral virtual polytopes. Then the function Φ on the lattice Zk defined by:

Φ(m1, . . . ,mk) =∑

x∈Zn

f(x)(β ∗ αm1

1 ∗ · · · ∗ αmk

k )(x),

where (m1, . . . ,mk) ∈ Zk, is the restriction of a polynomial on Rk of degree ≤ n+ d to Zk.

Finally we have the following variation of Corollary 4.8. We will use it later in Section5.5 and Section 5.8. Below Λ = 1

NZn ⊂ Rn for some ineteger N > 0.

Corollary 4.9. Let f : Rn → R be a polynomial of degree d and let γ1, . . . , γk ∈ P(Λ) bea k-tuple of Λ-virtual polytopes. Let a1, . . . , ak be fixed points in the lattice Λ and considerthe function Ψ : Zk → R defined by:

Ψ(m1, . . . ,mk) =∑

x∈(m1a1+···+mkak)+Zn

f(x)(γm1

1 ∗ · · · ∗ γmk

k )(x).

Then Ψ is the restriction of a polynomial on Rk to Zk.

Proof. We reduce the claim to Corollary 4.8. Consider the space Rn+k = Rn×Rk, the latticeΛ1 = 1

NZn+k ⊂ Rn+k and the Λ1-chains ρ1, . . . , ρk defined by ρi = γi ∗ δ(−ai,ei). Here ei is

the i-th standard basis vector in Rk and δ(a,b) denotes the characteristic function of a point

(a, b) ∈ Rn × Rk. Then ρm1

1 ∗ · · · ∗ ρmk

k = γm1

1 ∗ · · · ∗ γmk

k ∗ δ(−m1a1−···−mkak,m1e1+···+mkek).

Also consider the polynomial h : Rn×Rk → R defined by: h(x,m1, . . . ,mk) = f(x+a1m1+· · · + akmk). The claim now follows from Corollary 4.8 applied to the vector space Rn+k,the lattice Λ1 = 1

NZn+k ⊂ Rn+k, Λ1-chains ρ1, . . . , ρk and the polynomial h (we take β inCorollary 4.8 to be 0). �

5. Complete intersections in spherical varieties

In the rest of the paper we will use the following notation about reductive groups:

- G denotes a connected complex reductive algebraic group.- B a Borel subgroup of G and T , U the maximal torus and maximal unipotentsubgroups contained in B respectively.

- Λ is the weight lattice of G, Λ+ is the subset of dominant weights and ΛR = Λ⊗ZR.The cone generated by Λ+ is the positive Weyl chamber denoted by Λ+

R.

17

- Vλ denotes the irreducible G-module corresponding to a dominant weight λ. Alsovλ denotes a highest weight vector in Vλ.

- G/H denotes a spherical homogeneous space.- Λ′ = Λ(G/H) ⊂ Λ denotes the weight lattice of G/H , i.e. the sublattice of Λconsisting of weights of B-eigenfucntions in C(G/H).

5.1. Preliminaries on spherical varieties. A G-variety X is called spherical if a Borelsubgroup (and hence any Borel subgroup) has a dense orbit. If X is spherical it has afinite number of G-orbits as well as a finite number of B-orbits. Spherical varieties are ageneralization of toric varieties for actions of reductive groups. Analogous to toric varieties,the geometry of spherical varieties can be read off from associated convex polytopes andconvex cones. For a nice overview of the theory of spherical varieties we refer the reader to[Perrin14].

It is a well-known fact that if L is a G-linearized line bundle on a spherical variety then thespace of sections H0(X,L) is a multiplicity free G-module. For a quasi-projective G-varietyX this is actually equivalent to X being spherical.

Below are some important examples of spherical varieites and spherical homogeneousspaces:

(1) When G is a torus, the spherical G-varieties are exactly toric varieties.(2) The flag variety G/B and the parital flag varieties G/P are spherical G-varieties by

the Bruhat decomposition.(3) Let G × G act on G from left and right. Then the stabilizer of the identity is

Gdiag = {(g, g) | g ∈ G}. Thus G can be identified with the homogeneous space(G × G)/Gdiag. Again by the Bruhat decomposition this is a spherical (G × G)-homogeneous space.

(4) Consider the set Q of all smooth quadrics in Pn. The group G = PGL(n+1,C) actstransitively onQ. The stabilizer of the quadric x20+· · ·+x2n = 0 (in the homogeneouscoordinates) is H = PO(n+ 1,C) and hence Q can be identified with the homoge-neous space PGL(n+ 1,C)/PO(n+ 1,C) . The subgroup PO(n+ 1,C) is the fixedpoint set of the involution g 7→ (gt)−1 of G and hence Q is a symmetric homoge-neous space. In particular, Q is spherical. Let V be the vector space of all quadraticforms in n + 1 variables and V ∗ its dual. The map which assigns to a quadric Cits homogeneous equation (respectively equation of the dual quadric C∗ ) gives anembedding of Q in P(V ) (respectively P(V ∗)). Let X be the closure of the set of allquadrics (C,C∗) in P(V ) × P(V ∗). It is called the variety of complete quadrics. Itis well-known that X is a smooth variety (see [DeConcini-Procesi82, Theorem 3.1]).This variety plays an important role in classical enumerative geometry.

Throughout the rest of the paper we will fix a spherical homogeneous space G/H .

Definition 5.1. Let Λ(G/H) be the lattice of B-weights for the action of B on the fieldof rational functions C(G/H), i.e. the set of all λ ∈ Λ which appear as the weight of a B-eigenfunction in C(G/H). Clearly Λ(G/H) is a sublattice of Λ. We will denote the latticeof B-weights of G/H simply by Λ′.

Remark 5.2. Let C(G/H)(B) denote the multiplicative group of nonzero B-eigenfunctionsin C(G/H). If two B-eigenfunctions f and g have the same weight then f/g is a B-invariantrational function on G/H . Since X has an open B-orbit we conclude that f/g is constant.This proves that the map which sends a B-eigenfunction to its weight gives an isomorphismbetween C(G/H)(B)/C∗ and the lattice Λ′.

18

The following theorem about hp,0 numbers of spherical varieties will be used later. It isa generalization of Corollary 3.3 for toric varieties.

Theorem 5.3 (hp,0 numbers of spherical varieties). Let X be a smooth projective sphericalvariety. Then h0,0(X) = 1 and hp,0(X) = 0 if p > 0.

Proof. One shows that if X is a spherical variety then the maximal torus T has isolatedfixed points. The theorem then follows from the more general theorem that if X is a smoothprojective variety with an action of a torus with isolated fixed points then hp,q(X) = 0 if p 6=q. We refer to [Carrell-Liebermann73] for this vanishing result as well as its generalizationto holomorphic vector fields on Kahler manifolds. �

5.2. Cohomology of line bundles. LetX be an n-dimensional projective spherical varietywith a globally generated G-linearized line bundle L. The following result of Michel Briondetermines when the cohomology groups of the line bundles L and L−1 vanish ([Brion90]).It generalizes similar statements for toric varieties as well as the Borel-Weil-Bott theoremfor flag varieites. Below κ = κ(L) denotes the Itaka dimension of L, i.e. the dimension ofthe image of the Kodaira map of L⊗m for sufficiently large m.

Theorem 5.4 (Brion). Let X be a projective spherical variety with a globally generatedG-linearized line bundle L. Then:

(a) For any i > 0, Hi(X,L) = {0}.(b) For all i 6= κ, Hi(X,L−1) = {0}.

As before we let χ(X,L) denote the Euler characteristic of the bundle L defined byχ(X,L) =

∑ni=0(−1)idim(Hi(X,L)).

Corollary 5.5. Let X be a smooth projective spherical variety and L1, . . . ,Lk globally gen-erated G-linearized line bundles on X. For integers m1, . . . ,mk put:

ψ(m1, . . . ,mk) = χ(X,L⊗m1

1 ⊗ · · · ⊗ L⊗mk

k ).

Then ψ is a polynomial in the mi. Moreover when the mi are nonnegative we have:

ψ(m1, . . . ,mk) = dim(H0(X,L⊗m1

1 ⊗ · · · ⊗ L⊗mk

k )).

Proof. The corollary follows from the Hirzbruch-Riemann-Roch theorem and Theorem 5.4(a).�

In particular, let L be a globally generated G-linearized line bundle on a projectivespherical variety X . Let ψ be the polynomial in m ∈ Z such that ψ(m) = dim(H0(X,L⊗m))for any nonnegative integer m.

Corollary 5.6. As above let κ = κ(L) be the Itaka dimension of L. We can compute thedimension of Hκ(X,L−1) by:

dim(Hκ(X,L−1)) = (−1)κψ(−1).

(Note that by Theorem 5.4(b) all other cohomology groups of L−1 are 0.)

Proof. The corollary follows from the Hirzbruch-Riemann-Roch theorem and Theorem 5.4(b).�

19

5.3. Moment polytope. In this section we discuss the notion of the moment polytope∆(R) of a graded G-algebra R =

⊕

mRm. When R is the homogeneous coordinate ring ofa smooth projective G-variety (equivariantly embedded in a projective space) the momentpolytope can be identified with the moment polytope (or Kirwan polytope) of X in sym-plectic geometry (see Remark 5.8 below). We will be interested in the moment polytopesof graded G-linear systems R over a spherical variety X . When X is normal, the integralpoints in the dilated polytope m∆(R) correspond to the irreducible G-modules Vλ appear-ing in Rm (Theorem 5.9). We will express our main formula for the genus of a completeintersection in terms of integral points in moment polytopes (Theorem 5.29).

5.3.1. Moment polytope of a graded G-algebra. Let R =⊕

mRm be a graded G-algebra overC where the Rm are finite dimensional. Moreover, assume that R is contained in a finitelygenerated graded algebra. The rings R we will be interested in the upcoming sections areG-invariant graded linear systems on a spherical variety or a homogeneous space. That is,graded subalgebras of rings of sections of G-line bundles on a spherical variety.

Consider the additive semigroup S(R) ⊂ N× Λ defined by:

S(R) =⋃

m

{(m,λ) | Vλ appears in Rm}.

Let C(R) denote the closure of the convex hull of S(R) in the vector space R× ΛR. It is aclosed convex cone with apex at the origin. One defines the moment convex body ∆(A) tobe the slice of the cone C(R) at m = 1 ([Brion87] and [Kaveh-Khovanskii12b]). That is:

∆(R) = C(R) ∩ ({1} × ΛR).

Alternatively, after projection R×ΛR → ΛR on the second factor, the polytope ∆(R) canbe defined as:

∆(R) =⋃

m>0

{λ/m | Vλ appears in Rm}.

Remark 5.7. If the algebra R is finitely generated then one shows that the semigroupS(R) is a finitely generated semigroup and hence ∆(R) is a rational convex polytope. Inthis case we will refer to ∆(R) as the moment polytope. The moment polytope is also calledthe Brion polytope. It was first introduced in the paper [Brion87].

Remark 5.8 (Connection with moment polytope in symplectic geometry). Let K be acompact Lie group and let X be a compact Hamiltonian K-manifold with the moment mapφ : X → Lie(K)∗. It is a well-known result due to F. Kirwan that the intersection of theimage of the moment map with the positive Weyl chamber is a convex polytope usuallycalled the moment polytope or Kirwan polytope of the Hamiltonian K-space X .

As usual let G be a complex connected reductive group. Let G act linearly on a finitedimensional complex vector space V . Let X be a closed irreducible G-stable subvariety ofthe projective space P(V ). Let R denote the homogenous coordinate ring of X . Fix a K-invariant inner product on V where K is a maximal compact subgroup of G. This inducesa K-invariant symplectic structure on P(V ) and hence on the smooth locus of X .

(1) With this symplectic structure, the smooth locus of X is a Hamiltonian K-manifold.(2) When X is smooth it can be proved that the Kirwan polytope of X coincides with

∆(R) (see [Ness84], [Guillemin-Sternberg84] and [Brion87]). More precisely, theKirwan polytope identifies with ∆(R) after taking the involution λ 7→ λ∗, whereλ∗ = −w0λ.

20

(3) The above is still true ifX is non-smooth. In this case one considers the moment mapof P(V ) (as a Hamiltonian K-space) and restricts it to X . Then the intersection ofthe image of X (under the restricted moment map) with the positive Weyl chambercan be identified with ∆(R).

5.3.2. Moment polytope of a line bundle on a spherical variety. In this section we describethe linear inequalities defining the moment polytope of the ring of sections of a G-linearizedline bundle on a normal projective spherical variety. We will use it to describe the spaceof sections of the line bundle, as a G-module, in terms of the lattice points in its momentpolytope.

Let L be a G-linearized line bundle on a normal projective spherical G-variety X . ThenH0(X,L) is a finite dimensional G-module. Let us assume that H0(G,L) 6= {0}. By thering of sections of L we mean the algebra R(X,L) defined by:

R(X,L) =⊕

m

H0(X,L⊗m).

We denote the moment convex body of this algebra by ∆(X,L).Since H0(X,L) is a finite dimensional G-module there is a B-eigensection σ in H0(X,L).

Then the divisor D of the section σ is a B-stable divisor. Let D1, . . . , Ds be all the B-stableprime divisors in X . Thus we can write:

D =∑

i

aiDi.

Then for a rational function f ∈ C(X) the corresponding mermorphic section fσ belongsto H0(X,L) if and only if it satisfies:

ordDi(f) ≥ −ai, ∀i = 1, . . . , s,

where ordDiis the order of zero-pole along the prime divisor Di (notice that here we are

using the assumption that X is normal). Via restriction, the function ordDinow defines a

linear function ℓDion the lattice Λ′ = Λ(G/H) ∼= C(G/B)(B)/C∗ (see Remark 5.2). Let α

denote the weight of the B-eigensection σ ∈ H0(X,L) that we fixed. Then the G-spectrumof H0(X,L) can be described as:

(15) SpecG(H0(X,L)) = α+ {γ ∈ Λ′ | ℓDi

(γ) ≥ −ai, 1 ≤ i ≤ s},

= {λ ∈ α+ Λ′ | ℓDi(λ) ≥ −ai + ℓDi

(α), 1 ≤ i ≤ s}.

Applying this to all the H0(X,L⊗m) for m > 0, we get the following description of themoment polytope of the ring of sections R(X,L):

(16) ∆(X,L) = {x ∈ α+ Λ′R| ℓDi

(x) ≥ −ai + ℓDi(α), 1 ≤ i ≤ s}.

(Λ′Ris the vector space spanned by the lattice Λ′.)

Conversely, suppose λ = α + γ ∈ α + Λ′ is a shifted lattice point which lies in thepolytope ∆(X,L). By (16) for any i = 1, . . . , s we have ℓDi

(γ) ≥ −ai. And hence if f is aB-eigenfunction with weight γ then for any i we have ordDi

(f) ≥ −ai which implies thatfσ ∈ H0(X,L). Thus we have proved:

Theorem 5.9 (Brion).

H0(X,L) =⊕

λ∈∆(X,L)∩(α+Λ′)

Vλ.

21

Theorem 5.9 immediately gives us the dimension of the space of sections H0(X,L):

(17) dim(H0(X,L)) =∑

λ∈∆(X,L)∩(α+Λ′)

dim(Vλ)

For a rational polytope ∆ ⊂ ΛR and a ∈ Λ we define the number S(∆, a) by:

(18) S(∆, a) =∑

λ∈∆∩(a+Λ′)

f(λ),

where f is the Weyl polynomial given by the formula:

(19) f(λ) = dim(Vλ) =∏

α∈Φ+

〈λ + ρ, α〉/〈ρ, α〉.

By the Weyl dimension formula, for any dominant weight λ, the dimension of Vλ is equalto f(λ). With this notation in place, we can restate (17) as:

Proposition 5.10. For any interger m > 0 we have:

dim(H0(X,L⊗m)) = S(m∆(X,L),mα).

Next we consider the line bundle L−1. To describe the cohomology group Hκ(X,L−1),where κ = κ(L), we need some more notation. Let ∆ be a rational polytope in ΛR and a ∈ Λ.By Corollary 4.9, the function: Ψ(m) = S(m∆,ma) is a polynomial. We put S◦(∆, a) tobe (−1)dim(∆)+d∆Ψ(−1), where d∆ is the degree of the Weyl polynomial f restricted to theaffine span of ∆ in ΛR. In the light of Theorem 4.2(1), S◦(∆, a) can also be defined as:

(20) S◦(∆, a) = (−1)d∆

∑

λ∈∆◦∩(a+Λ′)

f(−λ).

As usual ∆◦ denotes the interior of ∆ (in the topology of the affine span of ∆).The following theorem determines the dimension of the cohomology group Hκ(X,L)

where κ = κ(L) is the Itaka dimension. We note that by Theorem 5.9 the Itaka dimensionκ = κ(L) is equal to dim(∆(X,L)) + d∆(X,L).

Theorem 5.11. dim(Hκ(X,L−1)) = S◦(∆(X,L), α).

Proof. By Corollary 4.9 we know that Ψ(m) = S(m∆(X,L),mα) is a polynomial in m.Moreover, by Corollary 5.5 we know that for any nonnegative integer m:

χ(X,L⊗m) = dim(H0(X,L⊗m)) = S(m∆(X,L),mα),

where χ(X,L⊗m) denotes the Euler characteristic of the bundle L⊗m. Putting these togetherwe conclude that dim(Hκ(X,L−1)) = (−1)κΨ(−1). Note that the polynomial χ(X,L⊗m)has degree κ which is equal to dim(∆) + d∆. This finishes the proof. �

5.4. Newton-Okounkov polytope. Fix a reduced decomposition w0 for the longest el-ement w0 in the Weyl group of G. To any λ in the positive Weyl chamber Λ+

Rone can

associate a convex rational polytope ∆w0(λ) ⊂ RN called the string polytope associated to

λ (and w0). Here N is the number of positive roots. The polytope ∆w0(λ) has the property

that, when λ is a dominant weight, the number of integral points in ∆w0(λ) is equal to the

dimension of the irreducible G-module Vλ, i.e.:

(21) dim(Vλ) = #(∆w0(λ) ∩ ZN ).

In fact, the integral points in ∆w0(λ) are in one-to-one correspondence with the so-called

canonical basis for Vλ ([Littelmann98, Berenstein-Zelevinsky01]).22

The string polytopes generalize the well-known Gelfand-Zetlin polytopes ∆GZ(λ) associ-ated to irreducible representations of GL(n,C) ([Gelfand-Zetlin50]). That is, for a specificchoice of a reduced decomposition, namely w0 = (s1)(s2s1)(s3s2s1)(snsn−1 · · · s1), for thelongest element of the Weyl group of GL(n,C) the string polytopes can be identified withthe Gelfand-Zetlin polytopes.

Remark 5.12. The dependence of ∆w0(λ) is piecewise linear, in the sense that there is a

rational polyhedral cone Cw0(with apex at the origin) in the vector space RN × ΛR such

that for each λ ∈ Λ+R

the string polytope ∆w0(λ) is the slice of the cone Cw

0at λ, i.e.

∆w0(λ) = Cw0

∩ π−1(λ) where π : RN × ΛR → ΛR is the projection on the second factor.Note that we can then define ∆w

0(λ) for all λ ∈ ΛR (not just integral λ).

Moreover, when G = GL(n,C) it is easy to see from the defining inequalities of theGelfand-Zetlin polytopes that λ 7→ ∆GZ(λ) is additive, namely for any λ, γ ∈ Λ+

Rwe have:

∆GZ(λ+ γ) = ∆GZ(λ) + ∆GZ(γ).

Now let X be a spherical G-variety with a G-linearized line bundle L. Let R =⊕

mRm

be a graded G-subalgebra of the ring of sections⊕

mH0(X,Lm). Moreover assume thatRm is finite dimensional for all m (this is automatic if X is projective). We call R a graded

G-linear system. We would like to associate a polytope ∆(R) to R which is responsible forthe dimensions of the homogeneous pieces Rm.

Definition 5.13. Let ∆(R) denote the Newton-Okounkov polytope of R, that is, the poly-tope over ∆(R) with string polytopes ∆w

0(λ) as fibers:

(22) ∆(R) =⋃

λ∈∆(R)

({λ} ×∆w0(λ)) ⊂ Λ+

R× RN .

If L is a G-linearized line bundle on a projective spherical G-variety X we denote theNewton-Okounkov polytope of the ring of sections R(X,L) by ∆(X,L).

Remark 5.14. The above notion of the Newton-Okounkov polytope of a graded G-linearsystem over a spherical variety is a special case of the more general notion of a Newton-Okounkov body of a graded linear system on an arbitrary variety (see [Lazarsfeld-Mustata09,Kaveh-Khovanskii12a] and the references therein).

Let L be a G-linearized line bundle on a normal projective spherical G-variety X whichcontains G/H as the open G-orbit. As usual Λ′ = Λ(G/H) denotes the lattice of weights ofB-eigenfunctions in C(G/H). Let us assume that H0(X,L) 6= {0} and fix a weight α of aB-eigensection in H0(X,L).

Let ∆ be a rational polytope in Rn. Fix a lattice L ⊂ Rn and a point a ∈ Rn. We denoteby N(∆, a) (respectively N◦(∆, a)) the number of points in the shifter lattice a+ L which

lie in ∆ (respectively in the interior of ∆ in the topology of its affine span). We note that,

by Theorem 4.2(1), N◦(∆, a) is equal to the (−1)dim(∆) times the value of the polynomial

m 7→ N(m∆,ma) at m = −1. These notations slightly extend N and N◦ defined in Section3.

Proposition 5.15 (Newton-Okounkov polytope and dimension of space of sections). Fixthe lattice Λ′ × ZN in the (real) vector space ΛR × RN . Consider α ∈ Λ as an element ofΛ× ZN . Then for any integer m > 0 we have:

dim(H0(X,L⊗m)) = N(m∆(X,L),mα).23

Moreover:

dim(Hκ(X,L−1)) = N◦(∆(X,L), α),

where as before κ = κ(L) is the Itaka dimension.

Proof. The proposition follows from Proposition 5.10 and Theorem 5.11. �

5.5. Euler characteristic of line bundles over a projective spherical variety. Let Xbe a projective spherical G-variety. In this section we consider the cases where the momentpolytope or the Newton-Okounkov polytope are additive. In these cases, extending the caseof toric varieties, the formula for Euler characteristic of line bundles on X can be expressedas polynomial measures on convex chains (Section 4.3).

Let L1, . . . ,Lk be G-linearized line bundles over a projective spherical variety X . Weassume that the bundles are globally generated (i.e. the H0(X,Li) are base point free linearsystems on X). Consider the bundle L⊗m1

1 ⊗ · · · ⊗ L⊗mk

k . Consider any collection of nonnegative numbers m1, . . . ,mk, such that at least one number is not zero. Below we willassume that one the following two conditions on these bundles hold:

(I) The Newton-Okounkov polytope ∆ of the bundle L⊗m1

1 ⊗· · ·⊗L⊗mk

k is equal to the

Minkowski sum ∆1+ · · ·+∆k where ∆1, . . . , ∆k are the Newton-Okounkov polytopeof the bundles L1, . . . ,Lk respectively.

(II) The moment polytope ∆ of the bundle L⊗m1

1 ⊗· · ·⊗L⊗mk

k is equal to the Minkowskisum ∆1 + · · · + ∆k where ∆1, . . . ,∆k are the moment polytopes of the bundlesL1, . . . ,Lk respectively.

Remark 5.16. (1) For k = 1 The Conditions I and II always hold but when k > 1 they donot always hold.

(2) The condition (II) above holds in the classes of horospherical homogeneous spacesas well as the group case (see Sections 6.1 and 6.2). The condition (I) holds in these twoclasses of examples provided that the string polytopes are also additive. This for examplehappens for the well-known Gelfand-Zetlin polytopes (which are a special case of the stringpolytopes for G = GL(n,C)).

Let as usual Λ′ denotes weight lattice of X , i.e. the sublattice of the weight lattice Λconsisting of all the weights of B-eigenfunctions on X . For each i = 1, . . . , k fix a weightai ∈ Λ of a B-eigensection in H0(X,Li).

Theorem 5.17. Assume that for the bundles L1, . . . ,Lk the condition (I) holds. Then forany k-tuple of integral numbers m1, . . . ,mk the Euler characteristic of X with the coefficientsin the bundle L⊗m1

1 ⊗ · · · ⊗ L⊗mk

k is given by:

(23) χ(X,L⊗m1

1 ⊗ · · · ⊗ L⊗mk

k ) =∑

x∈(m1a1+···+mkak)+(Λ′×ZN )

γm1

1 ∗ · · · ∗ γmk

k ,

where γi is the characteristic function of the Newton-Okounkov polytope ∆(X,Li).

Proof. By the Hirzbruch-Riemann-Roch theorem the Euler characteristic is a polynomialin the m1, . . . ,mk. Moreover by Proposition 5.15 it coincides with the righthand side of(23) for mi ≥ 0. On the other hand, by Corollary 4.9 the righthand side of (23) is also apolynomial in m1, . . . ,mk. Thus the two polynomials must coincide. �

Theorem 5.18. Assume that for the bundles L1, . . . ,Lk the condition (II) holds. Then forany k-tuple of integral numbers m1, . . . ,mk the Euler characteristic of X with the coefficients

24

in the bundle L⊗m1

1 ⊗ · · · ⊗ L⊗mk

k is given by:

(24) χ(X,L⊗m1

1 ⊗ · · · ⊗ L⊗mk

k ) =∑

x∈(m1a1+···+mkak)+Λ′

f(x)γm1

1 ∗ · · · ∗ γmk

k ,

where γi is the characteristic function of the moment polytope ∆(X,Li) and f is the Weylpolynomial (see (19)).

Proof. By the Hirzbruch-Riemann-Roch theorem the Euler characteristic is a polynomialin the m1, . . . ,mk. Also by Proposition 5.10 it coincides with the righthand side of (24)for mi ≥ 0. Again by Corollary 4.9 the righthand side is also polynomial and the twopolynomials must coincide. �

5.6. Invariant linear systems on a spherical homogeneous space. In this section wediscuss invariant linear systems on a spherical homogenous space G/H and their associatedmoment polytopes. First we consider the case of invariant subspaces of regular functionsand then the general case of invariant linear systems.

5.6.1. Moment polytope of an invariant subspace of regular functions. Let us assume thatthe homogeneous space G/H is quasi-affine. In this section we consider the case of trivialline bundles, i.e. we have finite dimensional subspaces of regular functions on G/H . Thissituation is closer to the classical toric case and Newton polytope theory which is concernedwith finite dimensional subspaces of Laurent polynomials spanned by monomials.

Example 5.19. (a) Let H = U be a maximal unipotent subgroup of G. One can showthat G/U is quasi-affine and as a G-module (for the left G-action on G/U) the spaceof regular functions C[G/U ] decomposes as:

C[G/U ] =⊕

λ∈Λ+

Vλ.

(b) Consider the left-right action of G×G on G. Then G ∼= (G×G)/Gdiag is affine andthe space of regular functions on G, as a (G×G)-module, decomposes as:

C[G] =⊕

λ∈Λ+

End(Vλ).

As usual let Λ′ = Λ(G/H) ⊂ Λ be the lattice of weights of B-eigenfunction in C(G/H).Also let Λ′+ = Λ+(G/H) denote the weights of the B-eigenfunctions in the algebra C[G/H ].It is clear that Λ′+ is a semigroup in Λ. One knows that the G-algebra C[G/H ] decomposesinto finite dimensional irreducible G-modules. Thus every B-eigenfunction in C[G/H ] isactually a highest weight vector for G and generates an irreducible G-module. Thus:

Λ′+ = {λ ∈ Λ+ | Vλ appear in C[G/H ]}.

The following is well-known (see [Timashev06]):

Proposition 5.20. The semigroup Λ′+ generates the lattice Λ′.

Now let A ⊂ Λ′+ be a finite subset. To A there corresponds the finite dimensionalG-invariant subspace:

LA =⊕

λ∈A

Vλ ⊂ C[G/H ].

Since LA is G-invariant it is automatically base point free and hence its Kodaira mapis defined everywhere on G/H . Let ΦA : G/H → P(L∗

A) denote the Kodaira map of theG-invariant subspace LA. It is a G-equivariant morphism. Also let YA denote the closure

25

of the image of G/H in the projective space P(L∗A). We note that the dimension of the

projective variety YA could possibly be smaller than that of G/H .

Let R(A) be the integral closure of the algebra R(A) =⊕

m LmA in

⊕

m C(G/H). Weknow that:

R(A) =⊕

m

LmA ,

where L denotes the integral closure (or completion) of a subspace L in the field of rationalfunctions C(G/H) (see [Samuel-Zariski60, Appendix 4]). We denote the moment polytopeof this graded algebra by ∆(A). It is a convex polytope containing A.

We have the following description of the G-modules LmA in terms of the moment polytope

∆(A):

Theorem 5.21. For every integer m > 0, the G-module LmA decomposes as:

LmA =

⊕

λ∈m∆(A)∩Λ′

Vλ.

Proof. As in the proof of Theorem 5.24, given the G-invariant subspace LA we can find anormal projective sphericalG-varietyX which contains G/H as the open orbit and moreoverthe Kodaira map ΦA extends to the whole X . Let L = Φ∗

A(O(1)) be the pull-back of theline bundle O(1) on the projective space to X . One shows that for each m > 0 the integralclosure Lm

A can be identified with H0(X,L⊗m). The theorem now follows from Theorem5.9. Note that α is the identity character because C[G/H ] contains the constant function 1which is invariant under the action of G and hence is a B-eigensection with weight 0. �

5.6.2. Moment polytope of an invariant linear system. Let E be a G-linearized line bundleon G/H . Then the space of sections H0(G/H, E) is a G-module. Moreover, since G/H isspherical, H0(G/H, E) is multiplicity-free, that is, every irreducible G-module appears in itwith multiplicity 0 or 1.

Take a finite nonempty subset A of the G-spectrum of the space of global sectionsH0(G/H, E). Let EA denote the G-invariant subspace of H0(G/H, E) determined by A,that is:

EA =⊕

λ∈A

Vλ.

Since EA is G-invariant then the base locus of EA, i.e. the locus of points where all thesections in EA vanish, is a G-invariant subvariety of G/H . But as EA is nonzero we concludethat the base locus of EA is empty, in other words, EA is base point free. The base pointfree linear system EA gives rise to a Kodaira map ΦA : X → P(E∗

A), the projective spaceof the dual space E∗

A. We denote the closure of the image of the Kodaira map by YA. TheKodaira map is G-equivariant and hence YA is a G-stable subvariety of the projective spaceP(E∗

A). Note that the dimension of the projective variety YA may be smaller than that ofX . To A we can associate a graded algebra:

R(A) =⊕

m

EmA ,

where EmA denotes the image of EA⊗· · ·⊗EA (m times) in H0(X, E⊗m) under the product

map H0(X, E) ⊗ · · · ⊗H0(X, E) (m times). The algebra R(A) is a graded G-subalgebra ofthe ring of sections

⊕

mH0(X, E⊗m). The homogneous coordinate ring of the projectivevariey YA ⊂ P(E∗

A) can naturally be identified with R(A).26

Let C(E) denote the space of meromorphic sections of E . It can be identified with C(G/H)via a choice of a nonzero section σ ∈ C(E). We denote the integral closure of the algebra

R(A) in⊕

m C(E⊗m) by R(A).We have the following extension of Theorem 5.21:

Theorem 5.22. For every integer m > 0, the G-module EmA decomposes as:

EmA =

⊕

λ∈m∆(A)∩(mα+Λ′)

Vλ.

Here α denotes the weight of a B-eigensection σ in H0(G/H, E).

Proof. As in the proof of Theorem 5.21. �

5.7. Simultaneous resolution of singularities. Let E1, . . . , Ek be globally generated G-linearized line bundles on a spherical homogeneous space G/H . For each i = 1, . . . , k letEi be a nonzero G-invariant linear system for Ei i.e. Ei is a finite dimensional G-invariantsubspace of H0(X, Ei). Each Ei is G-invariant and hence it is base point free. Thus theKodaira map ΦEi

is defined on the whole G/H . As usual we denote the closure of the imageof the Kodaira map ΦEi

by YEi. It is a projective G-subvariety of P(E∗

i ).We will be interested in a generic complete intersection of E1, . . . , Ek in G/H . To apply

topological methods, we need to work with a compactification of G/H which behaves nicelywith respects to the linear systems Ei. This is the content of the next definition. It is anextension of the similar notion for toric varieties ([Khovanskii77]).

Definition 5.23. Let X be a G-equivariant completion of G/H , that is, X is a completespherical G-variety which contains G/H as the open orbit. Let us say that X is sufficientlycomplete with respect to the linear systems E1, . . . , Ek if:

(a) X is smooth.(b) For each i = 1, . . . , k, the Kodaira map ΦEi

extends to a morphism on the wholeX .

Theorem 5.24. Given base point free G-invariant linear systems E1, . . . , Ek on G/Hthere exists a projective spherical variety X which is sufficiently complete with respect toE1, . . . , Ek.

Proof. One knows that the homogeneous space G/H has an equivariant projective comple-tion. Let Z be such a projective completion, i.e. Z is a projective spherical G-variety whichcontains G/H as the open orbit. Consider the map Φ : G/H → Z × YE1

× · · · × YEkgiven

by:Φ(x) = (x,ΦE1

(x), . . . ,ΦEk(x)),

and let Y be the closure of the image of Φ . The map Φ is a G-equivariant embedding andY is a projective subvariety of Z × YE1

× · · · × YEk. We note that Y contains G/H as an

open orbit and hence is a spherical G-variety. We have the following commutative diagram:

(25) G/H //

ΦE1···Ek

''PPPP

PPPP

PPPP

PP(E∗

1 )× · · · × P(E∗k)

��P((E1 · · ·Ek)

∗)

where the vertical arrow is the Segre map and E1 . . . Ek is the linear system which is theimage of E1⊗· · ·⊗Ek in H0(G/H, E1⊗· · ·⊗Ek). Thus in defining Y , instead of the Kodairamaps ΦE1

, . . . ,ΦEk, we alternatively could use the Kodaira map ΦE1···Ek

.27

The following theorem guarantees that Y has a G-equivariant resolution of singularities(see [Perrin14]).

Theorem 5.25 (Resolution of singularities for spherical varieties). Every spherical G-variety has a G-equivariant resolution of singularities.

Let π : X → Y be a G-equivariant resolution of singularities of Y as in the above theorem.For each i let πi be the projection Z × YE1

× · · · × YEk→ YEi

restricted to Y . For eachi = 1, . . . , k we have a commutative diagram:

(26) Xπ // Y

πi

��G/H

Φ

<<①①①①①①①①① ΦEi // YEi

Note that Φ (respectively π) is an isomorphisms from G/H (respectively the open orbit inX) to the open orbit in Y . Let us identify the homogeneous space G/H with the open orbit

in X via π−1 ◦Φ. Now for each i define ΦEito be πi ◦ π. Clearly ΦEi

is G-equivariant andextends the Kodaira map ΦEi

: G/H → YEi. This finishes the proof of the theorem. �

The following is a direct corollary of Thom’s transversality theorem (Theorem 2.9):

Theorem 5.26 (Transversality of generic hyperplane sections). As above let E1, . . . , Ek

be G-invariant nonzero linear systems on a spherical homogeneous space G/H and X asufficiently complete completion of G/H with respect to the Ei. For fi ∈ Ei let Hi =

{x ∈ G/H | fi(x) = 0} ⊂ X be the closure of the hypersurface defines by fi. Let fi ∈ Ei begeneric. We then have:

(1) For each i the hypersurface Hi ⊂ X is smooth.(2) For each i the hypersurace Hi intersects all the G-orbits in X transversely.(3) Either the intersection of the hypersurfaces H1, . . . , Hk is empty or they intersect

transversely.

5.8. Genus and hp,0 numbers of a complete intersection. As in Section 5.7 let E1, . . . , Ekbe a G-linearized line bundle on G/H . For each i = 1, . . . , k fix a dominant weight αi fora B-eigensection in H0(G/H, Ei). Also for each i = 1, . . . , k let Ai be a finite subset ofSpecG(H

0(G/H, Ei)) and:

Ei =⊕

λ∈Ai

Vλ ⊂ H0(G/H, Ei),

the corresponding G-invariant linear system. Also we denoted by ∆(Ei) the moment poly-

tope of the G-algebra⊕

mEmi .

By Theorem 5.24 we can find a sufficiently complete projective completion X of G/Hwith respect to E1, . . . , Ek. For each i = 1, . . . , k let Li = Φ∗

Ei(O(1)) be the pull-back of

the line bundle O(1) on the projective space P(E∗i ) to X . The line bundle Li is globally

generated because it is the pull-back of a globally generated line bundle.One shows that for any m > 0 the integral closure Em

i can be identified with the spaceof sections H0(X,L⊗m

i ). Hence the moment polytope ∆(X,Li) coincides with ∆(Ei), that

is the moment polytope of the integral closure⊕

mEmi (in particular the moment polytope

∆(X,Li) is independent of the sufficiently complete completion X).The next theorem gives a necessary and sufficient condition for a generic complete in-

tersection Xk from E1, . . . , Ek to be nonempty, in terms of the dimensions of the Newton-Okounkov or moment polytopes. It is a direct corollary of Theorems 2.14 and 2.19.

28

Theorem 5.27. A generic complete intersection Xk from E1, . . . , Ek is nonempty if andonly if E1, . . . , Ek are independent. That is, for any J ⊂ {1, . . . , k} we have dim(∆J ) ≥ |J |(equivalently dim(∆J ) + dJ ≥ |J |). Here ∆J (respectively ∆J) is the Newton-Okounkovpolytope (respectively the moment polytope) of the linear system EJ =

∏

i∈J Ei and dJ isthe degree of the Weyl polynomial restricted to the affine span of ∆J .

For the rest of the paper we assume that the condition in Theorem 5.27 is satisfied andhence a generic complete intersection Xk in nonempty.

Let us recall some notation. As in Proposition 5.15 fix the lattice Λ′ × ZN in the vectorspace ΛR×RN . Recall that for a rational polytope ∆ ⊂ ΛR×RN and a point a ∈ ΛR×ZN ,we denote by N(∆, a) (respectively N◦(∆, a)) the number of points in the shifted lattice

a+(Λ′×ZN ) which lie in ∆ (respectively in the interior of ∆). Moreover, we define N ′(∆, a)to be:

N ′(∆, a) = (−1)dim(∆)N◦(∆, a).

In fact, N ′(∆, a) is the value of the polynomial m 7→ N(m∆,ma) at m = −1. Similarly if ∆is a rational polytope in the vector space ΛR and α ∈ Λ a weight, we denote by S(∆, α) thesum of values of the Weyl polynomial f on the shifted lattice points α+ Λ′ which lie in ∆.Moreover, we denote by S′(∆, α) the value of the polynomial m 7→ S(m∆,mα) at m = −1.Using the notation introduced in the paragraph before Theorem 5.11 we can write:

S′(∆, α) = (−1)dim(∆)+d∆S◦(∆, α).

Here S◦(∆, α) is (−1)d∆ times the sum of values of the polynomial f(−λ) on the shiftedlattice points α+ Λ′ which lie in the interior of ∆.

Also recall that for each i = 1, . . . , k, αi ∈ Λ is the weight of a B-eigensection in Ei. Themoment polytope ∆(Ei) lies in the affine subspace αi+Λ′

R. We also consider αi as (αi, 0) in

the larger lattice Λ×ZN . The Newton-Okounkov polytope ∆(Ei) lies in the affine subspaceαi + (Λ′

R× RN ).

For each i, let κi be the Itaka dimension of the linear system Ei. From Proposition 5.10and Theorem 5.11, applied to the line bundles Li on X , we obtain the following:

Theorem 5.28. With notation as above we have:

(a)

dim(H0(X,Li)) = S(∆(Ei), αi) = N(∆(Ei), αi).

(b)

dim(Hκi(X,L−1i )) = S◦(∆(Ei), αi) = N◦(∆(Ei), αi).

In particular, the cohomology groups of (X,Li) and (X,L−1i ) only depend on the

Ei, i.e. are independent of the choice of the sufficiently complete completion X forG/H.

We now use the above to compute the genus of a complete intersection from E1, . . . , Ek inG/H . Let f1, . . . , fk be generic elements in E1, . . . , Ek respectively. For each i = 1, . . . , k letDi = {x ∈ G/H | fi(x) = 0} be the hypersurface defined by fi and let Xk = D1 ∩ · · · ∩Dk.

The next theorem is one of our main results.

Theorem 5.29 (Genus of a complete intersection in a spherical homogenous space). Withnotation as above we have:

(27) χ(Xk) = 1−∑

i1

S′(∆(Ei1 ), αi1 ) +∑

i1<i2

S′(∆(Ei1Ei2), αi1 + αi2)− · · ·

+ (−1)kS′(∆(E1 · · ·Ek), αi1 + · · ·+ αik).

29

or equivalently:

(28) χ(Xk) = 1−∑

i1

N ′(∆(Ei1), αi1 ) +∑

i1<i2

N ′(∆(Ei1Ei2), αi1 + αi2)− · · ·

+ (−1)kN ′(∆(E1 · · ·Ek), αi1 + · · ·+ αik).

Proof. The theorem follows from Theorem 2.20, Corollary 5.6, Theorem 5.11 and Proposi-tion 5.15. �

Remark 5.30. (1) When the moment polytope map E 7→ ∆(E) is additive, that is,∆(E1E2) = ∆(E1)+∆(E2) for any two G-linear systems E1, E2, then the formula (27) canbe computed only in terms of the polytopes ∆(Ei) (see Sections 6.1 and 6.2).

(2) Similarly, when the map E 7→ ∆(E) is additive, that is, ∆(E1E2) = ∆(E1) + ∆(E2)for any two G-linear systems E1, E2, then the formula (28) can be computed only in terms

of the polytopes ∆(Ei) (see Sections 6.1 and 6.2).

The following is the extension of Definition 3.8 to the spherical case:

Definition 5.31. Let E1, . . . , Ek be as above. We say that a nonnegative integer i iscritical for the E1, . . . , Ek if there is a nonempty set J ⊂ {1, . . . , k} such that N◦(∆J ) > 0

and dim(∆J )− |J | = i (equivalently dim(∆J) + dJ − |J | = i). Recall that ∆J (respectively∆J ) is the Newton-Okounkov polytope (respectively the moment polytope) of the linearsystem EJ =

∏

i∈J Ei.

Theorem 5.32 (hp,0 numbers of a complete intersection in a spherical homogeneous space).Let p be a nonnegative integer which is not critical for E1, . . . , Ek in the sense of Definition5.31. Then:

hp,0(Xk) =

{

1 p = 0

0 p 6= 0.

Proof. Follows from Theorem 5.3 and Theorem 2.21. �

Corollary 5.33. Suppose all the numbers 0 ≤ i < n − k are not critical for E1, . . . , Ek.In particular, this is the case if all the Newton-Okounkov polytopes ∆1, . . . , ∆k have fulldimension n = dim(G/H). Then all the numbers hi,0(Xk), 0 ≤ i ≤ n − k, are zero exceptfor h0,0(Xk) = 1 and hn−k,0(Xk) which can be computed from (27) or (28).

Proof. Under the assumptions in the corollary we have d(J) ≥ n − k and hence no p with1 ≤ p < n− k is a critical number and hence hp,0(Xk) = 0 by Theorem 5.32. �

6. Examples

6.1. Horospherical varieties. A subgroup H ⊂ G is called horospherical if it contains amaximal unipotent subgroup. The corresponding homogeneous space G/H is called horo-spherical homogeneous space. It can be shown that a horospherical homogeneous space isspherical, that is, it has an open B-orbit. A spherical G-variety X is horospherical if theopen G-orbit is horospherical.

It is well-known that:

Proposition 6.1. A subgroup H ⊂ G is horospherical if and only if there is a parabolicsubgroup P ⊂ G such that P ′ ⊂ H ⊂ P (see [Kaveh-Khovanskii11, Section 2.1] for a proof).

30

Example 6.2. Let U be a maximal unipotent subgroup of G. Then G/U is a quasi-affinehorospherical homogeneous space. The natural map G/U → G/B is a fibration over theflag variety G/B and the fibers are isomorphic to the maximal torus T . If G = GL(n,C),U can be taken to be the subgroup of upper-triangular matrices with 1’s on the diagonal.

Remark 6.3. The name horospherical comes from hyperbolic geometry (see [Timashev06,Example 7.1]).

Let λ ∈ Λ+. For a rational G-module M let us denote the λ-isotypic component ofM byMλ, that is, Mλ is the sum of all the copies of the irreducible G-module Vλ in M . Clearly,if M is multiplicity free then Mλ is either {0} or isomorphic to Vλ. In particular, if E isa G-linearized line bundle over a spherical homogeneous space G/H then H0(G/H, E)λ iseither {0} or Vλ. The following is well-known (see [Popov86]):

Theorem 6.4. Let G/H be a horospherical homogeneous space. Let E1 and E2 be G-linearized line bundles on G/H with Ei ⊂ H0(G/H, Ei), i = 1, 2, two G-invariant linearsystems. Then for any λ, γ ∈ Λ the product (E1)λ(E2)γ lies in (E1E2)λ+γ , where E1E2 isthe linear system in E1⊗E2 spanned by all the products f1f2, fi ∈ Ei, and (E1E2)λ+γ is the(λ+ γ)-isotypic component of E1E2.

Let E be a G-linearized line bundle on G/H and take a finite subset A:

A ⊂ SpecG(H0(G/H, E)) = {λ | Vλ appears in H0(G/H, E)}.

As before we let EA denote the linear system:

EA =⊕

λ∈A

Vλ.

As in [Kaveh-Khovanskii11] one shows that:

Corollary 6.5. With notation as above, the moment polytope of the graded algebra⊕

mEmA

coincides with the convex hull of A.

Definition 6.6. For each finite subset A ⊂ Λ let ∆(A) denote the convex hull of A. In fact∆(A) is the moment polytope ∆(EA) of its associated linear system. Also

∆(A) =⋃

λ∈∆(A)

{λ} ×∆w0(λ),

the corresponding Newton-Okounkov polytope. Recall that w0 is a fixed reduced decom-position for the longest element w0 in the Weyl group W and ∆w0

(λ) ⊂ RN is the stringpolytope associated to λ and w0 (see Section 5.4).

From Theorem 6.4 it follows that for any two finite subsets A1,A2 we have EA1EA2

=EA1+A2

. As in [Kaveh-Khovanskii11] we have the following:

Proposition 6.7 (Additivity of the moment and Newton-Okounkov polytopes). (i) Themap E 7→ ∆(E), which associates to an invariant linear system E its moment poly-tope, is additive. This is basically the condition (II) in Section 5.5.

(ii) Suppose the string polytope map λ 7→ ∆w0(λ) is additive (e.g. G = GL(n,C) and

Gelfand-Zetlin polytopes). Then the map E 7→ ∆(E) is also additive. This isbasically the condition (I) in Section 5.5.

31

Let E1, . . . , Ek be G-linearized line bundles. For each i = 1, . . . , k let Ai be a finite subsetof SpecG(H

0(G/H, Ei)) ⊂ Λ+ and Ei ⊂ H0(G/H, Ei) the corresponding G-invariant linearsystem. Also to simplify the notation, for each i let ∆i denote the convex polytope ∆(Ai),i.e. the convex hull of the finite set Ai.

Corollary 6.8 (Genus of a complete intersection in a horospherical homogenous space).Let f1, . . . , fk be generic elements in E1, . . . , Ek respectively. For each i = 1, . . . , k letDi = {x ∈ G/H | fi(x) = 0} be the hypersurface defined by fi and let Xk = D1 ∩ · · · ∩Dk.Then:

(29) χ(Xk) = 1−∑

i1

S′(∆i1 , αi1) +∑

i1<i2

S′(∆i1 +∆i2 , αi1 + αi2)− · · ·

+ (−1)kS′(∆1 + · · ·+∆k, α1 + · · ·+ αk).

Remark 6.9. As in Remark 5.30(2) if the string polytopes are additive for the choice ofthe reduced decomposition w0 (e.g. G = GL(n,C) and Gelfand-Zetlin polytopes) then the

Newton-Okounkov polytope is also additive. Let ∆i denote the Newton-Okounkov polytope∆(Ai) associated to the finite subset Ai. Then, under the assumption of additivity of thestring polytopes, we can rewrite the formula (29) for the genus in terms of the Newton-

Okounkov polytopes ∆i and their sums:

(30) χ(Xk) = 1−∑

i1

N ′(∆i1 , αi1) +∑

i1<i2

N ′(∆i1 + ∆i2 , αi1 + αi2)− · · ·

+ (−1)kN ′(∆1 + · · ·+ ∆k, α1 + · · ·+ αk).

In particular if G/H is a quasi-affine variety we can consider the finite G-invariant subsetsof the ring of regular functions C[G/H ], that is, when we take all the line bundles Ei to bethe trivial line bundle. In fact, as we now explain, to each face of the Weyl chamber Λ+

Rthere

corresponds a quasi-affine horospherical homogeneous space and by Proposition 6.1 everyhorospherical homogeneous space G/H is a quotient of such a quasi-affine horospherical ho-mogeneous space: Let σ be a face of the Weyl chamber Λ+

R. Let P denote the corresponding

parabolic subgroup. When G = GL(n,C), the parabolic subgroups (up to conjugation)are block upper triangular subgroups. Let P ′ denote the commutator subgroup of P . Oneshows that the homogeneous space G/P ′ is quasi-affine and moreover, the algebra of regularfunctions C[G/P ′], for the natural action of G, decomposes as follows ([Popov-Vinberg72]):

C[G/P ′] =⊕

λ∈σ∩Λ+

V ∗λ .

Now each finite subset A ⊂ σ ∩ Λ+ determines a G-invariant subspace LA =⊕

λ∈A Vλ ofC[G/P ′]. Let us take finite subsets A1, . . . ,Ak of the semigroup σ ∩ Λ. Then Corollary6.8 gives us a formula for the genus of a generic complete intersection from the subspacesL1, . . . , Lk in the quasi-affine variety G/P ′, in terms of the convex hulls ∆i of the subsetsAi.

Finally, below is a concrete example of Corollary 6.8.

Example 6.10. Let V be a finite dimensional G-module. Let v1, . . . , vs be highest weightvectors of V with highest weights λ1, . . . , λs respectively. Put v = v1 + · · · + vs and letX be the closure of the G-orbit of v in V . It is an affine horospherical subvariety of V .Let L be the linear subspace of C[X ] consisting of linear functions in V ∗ restricted to X .As above one observes that the moment polytope ∆ of the subspace L, or equivalently the

32

graded algebra⊕

m Lm, is ∆ = conv{λi | i = 1, . . . , s}. Also let ∆ denote the correspondingNewton-Okounkov polytope. Then if f is a generic element in L defining a hypersurfaceHf = {x ∈ X | f(x) = 0}, Corollary 6.8 implies that the genus of Hf is equal to S◦(∆) =

N◦(∆), i.e. the number of integral points in the interior of the polytope ∆.

6.2. Group embeddings. Let π : G→ GL(n,C) ⊂ Mat(V ) be a finite dimensional repre-sentation of a connected reductive group G. Let πij : G → C, i, j = 1, . . . , n be the matrixelements, i.e. the entries of π. Let Lπ be the subspace of regular functions on G spanned bythe πij . Consider the action of G ×G on G by the multiplication from left and right. Thesubspace Lπ is a (G×G)-invariant subspace. Let Λ+

R(respectively W ) denote the positive

Weyl chamber (respectively the Weyl group) of G.

Definition 6.11 (Weight polytope). The convex hull of the Weyl orbit of the highestweights of the representation π is called the weight polytope of π. We will denote the weightpolytope by Pπ and its intersection with the positive Weyl chamber by P+

π .

As in [Kazarnovskii87], one can show that:

Theorem 6.12. The moment polytope of the (G×G)-algebra⊕

m Lmπ (which lives in Λ+

R×

Λ+R) can be identified with P+

π . To identify the moment polytope and the polytope P+π we

should send a point (λ, λ∗) to λ.

Let F denote the Weyl polynomial for the group G × G. It is a polynomial on thevector space ΛR × ΛR (see Section 5.3, paragraph after Theorem 5.9 for the definition ofthe Weyl polynomial). For any dominant weight of G × G of the form (λ, λ∗) we haveF (λ, λ∗) = dim(Vλ × V ∗

λ ) = dim(Vλ)2.

Let us take k representation π1, . . . , πk where k ≤ dim(G). For each πi let Li (respectivelyP+i ) be its subspace of matrix elements (respectively its weight polytope intersected with

the positive Weyl chamber).

Corollary 6.13 (Genus of a complete intersection in a group). Let f1, . . . , fk be genericelements in Lπ1

, . . . , Lπkrespectively. For each i = 1, . . . , k let Di = {x ∈ G/H | f(x) = 0}.

and let Xk = D1 ∩ · · · ∩Dk. Then:

(31) χ(Xk) = 1−∑

i1

S′(P+1 ) +

∑

i1<i2

S′(P+1 + P+

2 )− · · ·

+ (−1)kS′(P+1 + · · ·P+

k ).

As usual to a representation π we can associate a Newton-Okounkov polytope ∆π definedby:

∆π =⋃

λ∈P+π

({(λ, λ∗)} ×∆w0(λ)×∆w

0(λ∗)).

As in [Kaveh-Khovanskii10] we have the following:

Proposition 6.14 (Additivity of the moment and Newton-Okounkov polytopes). (i)The map π 7→ P+

π , which associates to a representation π its polytope P+π , is additive

with respect to the tensor product of representations. This is basically the condition(II) in Section 5.5.

(ii) Suppose the string polytope map λ 7→ ∆w0(λ) is additive (e.g. G = GL(n,C) and the

Gelfand-Zetlin polytopes). Then the map π 7→ ∆π is also additive. This is basicallythe condition (I) in Section 5.5.

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As in (ii) above, suppose the string polytope map is additive. Let ∆i denote the Newton-Okounkov polytope of the representaiton πi in Corollary 6.13. Then we can rewrite theformula (31) for the genus in terms of the Newton-Okounkov polytopes ∆i and their sums:

(32) χ(Xk) = 1−∑

i1

N ′(∆i) +∑

i1<i2

N ′(∆i1 + ∆i2))− · · ·

+ (−1)kN ′(∆1 + · · ·+ ∆k).

6.3. Flag varieties. Let X = G/B be the complete flag variety of a connected complexreductive algebraic group.

Corollary 6.15 (Genus of a complete intersection in the flag variety). Let λ1, . . . , λk bedominant weights. Let D1, . . . , Dk be smooth and transversely intersecting divisors for thecorresponding line bundles Lλ1

, . . . , Lλk. Let Xk = D1 ∩ · · · ∩Dk. Then:

(33) χ(Xk) = 1−∑

i1

N ′(∆w0(λi1 )) +

∑

i1<i2

N ′(∆w0(λi1 + λi2 ))− · · ·

+ (−1)kN ′(∆w0(λ1 + · · ·+ λk)).

Finally we briefly discuss the case of the flag variety of G = GL(n,C) and Gelfand-Zetlinpolytopes. Let G = GL(n,C). The flag variety of G can be identified with the variety of allflags of linear subspace in Cn:

{0} $ F1 $ · · · $ Fn = Cn.

Each dominant weight λ of G can be represented as an increasing n-tuple of integers:

λ = (λ1 ≤ · · · ≤ λn).

In their well-known work [Gelfand-Zetlin50], given a dominant weight λ, Gelfand and Zetlinconstruct a natural vector basis for the irreducible representation Vλ whose elements areparameterized with the integral points xi,j satisfying the following set of interlacing inequal-ities:

(34)

λ1 λ2 · · · · · · · · · λn

x1,n−1 x2,n−1 · · · · · · xn−1,n−1

x2,n−2 x2,n−2 · · · xn−2,n−2

· · · · · · · · ·

x1,2 x2,2

x1,1

where the notationa b

c

means a ≤ c ≤ b.The set of all points (xi,j) in Rn(n−1)/2 satisfying (34) is called the Gelfand-Zetlin poly-

tope associated to λ denoted by ∆GZ(λ). Let Lλ be the G-line bundle on the flag varietyassociated to a dominant weight λ. If H is a generic divisor of Lλ then Corollary 6.15states that the genus of H is equal to the number of integral points in Rn(n−1)/2 lying in

34

the interior of ∆GZ(λ). In other words, the number of integral points in Rn(n−1)/2 whichsatisfy the inequalities in (34) where all the inequalities are strict.

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Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, USA.

E-mail address: kaveh@pitt.edu

Department of Mathematics, University of Toronto, Toronto, Canada; Moscow Independent

University, Moscow, Russia.

E-mail address: askold@math.utoronto.ca

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